THE CONTACT PROPERTY
FOR MAGNETIC FLOWS ON SURFACES
Gabriele Benedetti
Trinity College
and
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
July 2014
A David e Marisa
This dissertation is the result of my own work and includes nothing
that is the outcome of work done in collaboration except where
specifically indicated in the text.
This dissertation is not substantially the same as any that I
have submitted for a degree or diploma or any other qualification
at any other university.
Gabriele Benedetti
October 18, 2014
The contact property for magnetic flows on surfaces
Gabriele Benedetti
Summary
This work investigates the dynamics of magnetic flows on closed orientable Rie
mannian surfaces. These flows are determined by triples (M, g, σ), where M is the
surface, g is the metric and σ is a 2form on M . Such dynamical systems are de
scribed by the Hamiltonian equations of a function E on the tangent bundle TM
endowed with a symplectic form ωσ, where E is the kinetic energy. Our main goal
is to prove existence results for
a) periodic orbits and b) Poincare´ sections
for motions on a fixed energy level Σm := {E = m2/2} ⊂ TM .
We tackle this problem by studying the contact geometry of the level set Σm.
This will allow us to
a) count periodic orbits using algebraic invariants such as the Symplectic Cohomol
ogy SH of the sublevels ({E ≤ m2/2}, ωσ);
b) find Poincare´ sections starting from pseudoholomorphic foliations, using the
techniques developed by Hofer, Wysocki and Zehnder in 1998.
In Chapter 3 we give a proof of the invariance of SH under deformation in an
abstract setting, suitable for the applications.
In Chapter 4 we present some new results on the energy values of contact type.
First, we give explicit examples of exact magnetic systems on T2 which are of contact
type at the strict critical value. Then, we analyse the case of nonexact systems on
M 6= T2 and prove that, for large m and for small m with symplectic σ, Σm is of
contact type. Finally, we compute SH in all cases where Σm is convex.
On the other hand, we are also interested in nonexact examples where the
contact property fails. While for surfaces of genus at least two, there is always a
level not of contact type for topological reasons, this is not true anymore for S2. In
Chapter 5, after developing the theory of magnetic flows on surfaces of revolution,
we exhibit the first example on S2 of an energy level not of contact type. We also
give a numerical algorithm to check the contact property when the level has positive
magnetic curvature.
In Chapter 7 we restrict the attention to low energy levels on S2 with a symplectic
σ and we show that these levels are of dynamically convex contact type. Hence, we
prove that, in the nondegenerate case, there exists a Poincare´ section of disctype
and at least an elliptic periodic orbit. In the general case, we show that there are
either 2 or infinitely many periodic orbits on Σm and that we can divide the periodic
orbits in two distinguished classes, short and long, depending on their period. Then,
we look at the case of surfaces of revolution, where we give a sufficient condition for
the existence of infinitely many periodic orbits. Finally, we discuss a generalisation
of dynamical convexity introduced recently by Abreu and Macarini, which applies
also to surfaces with genus at least two.
Acknowledgements
Sabra`s que no te amo y que te amo
puesto que de dos modos es la vida,
la palabra es un ala del silencio,
el fuego tiene una mitad de fr´ıo.
Pablo Neruda, Soneto XLIV
I express my deep gratitude to the following people, things and places.
To my advisor Gabriel Paternain, for many helpful discussions about the dy
namics of the magnetic field and for his invaluable support in preparing this thesis.
To Peter Albers and Ivan Smith, for kindly agreeing to be the examiners of this
thesis.
To Alex Ritter, for several conversations that have been fundamental to develop
the material contained in Chapter 3.
To Jungsoo Kang, for a nice conversation about multiplicity results for Reeb
orbits using equivariant Symplectic Cohomology.
To Leonardo Macarini, for pointing out to me a mistake in Inequality (7.27)
contained in a previous version of this work.
To Viktor Ginzburg, for his helpful and kind comments on Proposition F.
To Marco Golla, for the simple and nice proof about the parity of the linking
number contained in Lemma 6.11 and for his genuine interest in my work.
To Joe Waldron, for sharing office and food with me every day for a year.
To Jan Sbierski, for squash, badminton and basketball and a memorable formal
dinner at Magdalene with Paris Breast served for dessert.
To Paul Wedrich, for the delicious lunches at Churchill.
To Christian Lund, for coming to my office every day after lunch for pushups
training and for bringing me from a series of 5 to a series of 20 in one month.
To Nick ShepherdBarron, because his disembodied and charming voice coming
from the first floor has been a familiar presence for all the offices on the ground
floor.
To the neighbouring Office E0.19, for all the moments of inspiration.
To Carmelo Di Natale, for his mighty hugs and his sincere blessings.
To Enrico Ghiorzi, for popping up in my office to say “Hello!” at the end of the
working day with the helmet already worn.
To Volker Schlue, for sending from Toronto a correct proof that 0 = 1 and very
accurate weather forecasts written in Korean.
To Peter Herbrich, for his crucial help during my first days in Cambridge, for
bringing me with his red Polo around Scotland on a hiking trip in spite of rain
and blisters and for inviting me to the Polterabend and the fabulous wedding with
Melina (“Cuscino!”) in Lunow.
To Gareth, for helping me hiking my way through the PhD, from the green and
wet valleys of the Isle of Skye to the windy and rocky peak of the Innerdalst˚arnet,
and for reminding me that “Power is nothing without control”.
To John Christian Ottem, for all the good music, for climbing up a mountain
only to rescue a plastic bag I thought I had left on top (it was in my backpack, in
the end!), for still believing that I can arrive on time, for the beautiful drawing on
the blackboard and for all the mathematical jokes and puzzles.
To Anna Gurevich, for the dancing in Lunow and the nice chats about feminism
and particles moving on spheres.
To Achilleas Kryftis, for all the runs we went for together, and for teaching
me Greek in the meanwhile. I will miss the three llamas in Coton, the hill and a
rocambolesque escape from the shooting range.
To Christina Vasilakopoulou, for waving at me from the window every day as
I entered Pavilion E from the back, for the movienights full of “nibbly things” she
has organised with Mitsosat their place.
To the inhabitants of CMS, present, past and future. Especially to Li Yan, Guido
Franchetti, Micha l Przykucki, Tamara Von Glehn, Guilherme Frederico Lima, Ha
Thu Nguyen, Sebastian Koch, Kirsty Wan, Kyriakos Leptos, Bhargav Narayanan,
Gunnar Peng, Rahul Jha, Ashok and Arun Thillaisundaram, Cangxiong Chen,
Shoham Letzter, Dmitry Tonkonog, Claudius Zibrowius, Tom Gillespie, Vittoria
Silvestri and Sara Ricco`.
To all the students I supervised in Cambridge, for their energy and their patience.
To Elena Giusti, because now she knows why the prime numbers are infinite.
To Thomas Bugno, and his sheer passion for knowledge. I am sorry that he got
really upset to find out that 0! = 1.
To Laura Moudarres, for her excellent potato bake and for sharing with me the
passion for the salad with apple and cumin.
To Barbara Del Giovane, because she never talked about the weather, but pre
ferred to listen to my doubts and speak about hers with naturalness and empathy
To Andrew Chen, for making me proofread all the very formal messages he sent
to Italian libraries for getting access to ancient books.
To Francesca Pariti, because her smile and good mood has been the perfect
substitute for the English sun.
To Flavia Licciardello, for making me feel accepted and independent and an
adult.
To Daniele Dorigoni, for his extraordinary generosity counterbalanced by his
punting skills.
To Marco Mazzucchelli and Ana Lecuona, for all the wise advice about the future
and for making my stay in Marseille the most heartwarming experience.
To Alfonso Sorrentino, for being a wonderful mentor and dear friend in my life
and for bringing me to eat the best hummus on Earth.
To Giulio Codogni, because he is not afraid of asking “Can I don’t do the English
test?”. He has always been an example to me and the protagonist of the funniest
moments in Cambridge. Cheers a lot!
To Fano “bello da vicino e pure da lontano”, for being a really exquisite person
and for having organised the best Mercante in Fiera of my life.
To Luca Sbordone, because he is the most talented and entertaining storyteller.
It doesn’t make much difference to me whether what he says really happened or not.
To Ugo Siciliani De Cumis, who shared with me the darkest hours, because he
can cook amazing food (and suffer mine), fix all sort of things (which I had broken,
of course) and build a mixer between the cold and hot tap.
To the thirtyfive guests of 24 St. Davids House (Lin, you will be soon the
thirtysixth, last and longwaited!), for making this place a home to me.
To Mara Barucco and Simone Di Marino, for getting married.
To Pisa, Scuola Normale and Collegio Faedo, for all the magic encounters and
because a part of me (as well as my shoes) has remained there in these three years.
To Pietro Vertechi, for his genuine hospitality in Lisbon and for chasing a bar
up the stairs of a multistorey car park.
To Valerio Melani and Maria Beatrice Pozzetti, for staying connected in spite of
the distance.
To Anna Belle´, for bringing me the most unexpected schiacciatine of my entire
life when I was ill in bed.
To Claudia Flandoli, for visiting with me the cold and empty rooms of Palazzo
Ducale in Urbino with a pair of crutches e “un sorriso coinvolgente”.
To Alessandro Di Sciullo, Marco Cucchi, Jacopo Preziosi Standoli, Luca Grossi,
Paolo Magnani, Luca Borelli and Ermanno Finotti, for countless goodbye dinners.
To Claudio Baglioni, Alessandro Mannarino, Giorgio Gaber, Samuele Bersani,
Tool, Pain of Salvation, Banco del Mutuo Soccorso, Pink Floyd, Rachmaninov for
being the soundtrack of these years.
To Il Nido Del Cuculo, for all the silly laughs in front of the screen.
To Psychodynamic therapy, for making me discover a part of me, which I didn’t
know.
To Cognitive Behavioural Therapy, for reminding me of a part of me that I know
too well.
To Margherita, Eleonora, Andrea, Federica, Marisa, David, Agnese, Delvina,
Placido and Riccardo for all their love: it has been shaped in the form of a thesis.
Malo mortuum impendere quam vivum occidere.
Petronius, Satyricon
Then there is an intellectual or a student or two, very few
of them though, here and there, with ideas in their heads
that are often vague or twisted. Their “country” consists of
words, or at the most of some books. But as they fight they
find that those words of theirs no longer have any meaning,
and they make new discoveries about men’s struggles, and
they just fight on without asking themselves questions, until
they find new words and rediscover the old ones, changed
now, with unsuspected meanings.
Italo Calvino, The Path to the Spiders’ Nests
Every two or three days, at the moment of the check, he
told me: ”I’ve finished that book. Have you another one to
lend me? But not in Russian: you know that I have dif
ficulty with Russian.” Not that he was a polyglot: in fact,
he was practically illiterate. But he still ‘read’ every book,
from the first line to the last, identifying the individual let
ters with satisfaction, pronouncing them with his lips and
laboriously reconstructing the words without bothering about
their meaning. That was enough for him as, on different
levels, others take pleasure in solving crossword puzzles, or
integrating differential equations or calculating the orbits of
the asteroids.
Primo Levi, The Truce
Contents
Chapter 1. Introduction 1
1.1. Overview of the problem 1
1.2. Contact hypersurfaces and Symplectic Cohomology 2
1.3. The contact property on surfaces of revolution 6
1.4. Low energy levels of contact type on the twosphere 7
Chapter 2. Preliminaries 13
2.1. General notation 13
2.2. The geometry of an oriented Riemannian surface 14
2.3. Magnetic fields 17
2.4. Hamiltonian structures 23
Chapter 3. Symplectic Cohomology and deformations 27
3.1. Convex symplectic manifolds 27
3.2. Symplectic Cohomology of convex symplectic manifolds 29
3.3. Perturbing a nondegenerate autonomous Hamiltonian 33
3.4. Reeb orbits and two filtrations of the Floer Complex 34
3.5. Invariance under isomorphism and rescaling 40
3.6. Invariance under deformations projectively constant on tori 40
Chapter 4. Energy levels of contact type 53
4.1. Contact property for case (E) 56
4.2. Contact property for case (NE) 59
4.3. Symplectic Cohomology of a round sphere 62
4.4. Lower bound on the number of periodic orbits 64
Chapter 5. Rotationally symmetric magnetic systems on S2 69
5.1. The geometry of a surface of revolution 69
5.2. Estimating the set of energy levels of contact type 70
5.3. The symplectic reduction 74
5.4. Action of ergodic measures 78
Chapter 6. Dynamically convex Hamiltonian structures 79
6.1. Poincare´ sections 79
6.2. Elliptic periodic orbits 82
Chapter 7. Low energy levels of symplectic magnetic flows on S2 85
xiii
7.1. Contact forms on low energy levels 85
7.2. The expansion of the Ginzburg action function 89
7.3. Contactomorphism with a convex hypersurface 92
7.4. A direct estimate of the index 95
7.5. A dichotomy between short and long orbits 100
7.6. A twist theorem for surfaces of revolution 103
Bibliography 111
CHAPTER 1
Introduction
1.1. Overview of the problem
Let the triple (M, g, σ) represent a magnetic system, where M is a closed man
ifold, g is a Riemannian metric and σ is a closed 2form on M . We say that a
magnetic system is exact, nonexact or symplectic if σ is such. Such data give rise
to a Hamiltonian vector field XσE on the symplectic manifold (TM,ωσ := dλ−pi∗σ),
where pi is the footpoint projection from TM to M and λ is the pullback of the
Liouville 1form on the cotangent bundle via the duality isomorphism given by g.
The kinetic energy E(x, v) = 12gx(v, v) is the Hamiltonian function associated to
XσE . These systems model the motion of a charged particle on the manifold M
under the influence of a magnetic force, which enters the evolution equations via
the term −pi∗σ in the symplectic form. In the absence of a magnetic force, namely
when σ = 0, the particle is free to move and we get back the geodesic flow of the
Riemannian manifold (M, g). The zero section is the set of rest points for the flow
and all the smooth hypersurfaces Σm := {E = m22 }, with m > 0, are invariant
sets. Hence, we can analyse the vector fields XσE
∣∣
Σm
separately for every m and
compare the dynamics for different value of such parameter. In the geodesic case,
no interesting phenomenon arises from this point of view. Even if the dynamics of
the geodesic flow has a very rich structure and its understanding requires a deep
study, the flows on Σm and Σm′ are conjugated up to a constant time factor and,
therefore, the parameter m does not play any role in the analysis.
On the other hand, the situation for a general magnetic term σ is quite dif
ferent. As the parameter m varies and passes through some distinguished values,
the dynamics on Σm can undergo significant transformations, in the same way as
the topology of the levels of a Morse function changes when we cross a critical
value [CMP04]. How do we detect and measure these differences? Since the flows
ΦX
σ
E
∣∣
Σm
can exhibit a very complicated behaviour, we have to focus our analysis on
some simple property of these systems. In the present work we have chosen to look
at the periodic orbits. In particular, we examine
a) existence and multiplicity of periodic orbits on a given free homotopy class;
b) knottedness of a single periodic orbit and linking of pairs, through the existence
of global Poincare´ sections and periodic points of the associated return map;
c) local properties of the flow at a periodic orbit (e.g. stability), that are related to
the linearisation of the flow (elliptic/hyperbolic orbits).
Periodic orbits of magnetic flows received much attention in the last thirty years and
here we seize the occasion to mention some of the relevant literature on the subject.
1
2 1. INTRODUCTION
When M is a surface, the two classical approaches that have been pursued
are MorseNovikov theory and symplectic topology (see Ta˘ımanov’s [Ta˘ı92] and
Ginzburg’s [Gin96] surveys for details and further references). Refinements of these
old techniques and completely new strategies have been developed recently. Some au
thors work with (weakly) exact magnetic forms [BT98, Pol98, Mac04, CMP04,
Osu05, Con06, Pat06, FS07, Mer10, Ta˘ı10, AMP13, AMMP14, AB14].
Others seek periodic solutions with low kinetic energy [Sch06], the majority of
them assuming further that σ is symplectic [Ker99, GK99, GK02a, Mac03,
CGK04, GG04, Ker05, Lu06, GG09, Ush09]. Schneider’s approach [Sch11,
Sch12a, Sch12b] for orientable surfaces and symplectic σ uses a suitable index
theory for vector fields on a space of loops and shows in a very transparent way
how the Riemannian geometry of g influences the problem. Finally, we point out
[Koh09] where heat flow techniques are employed and [Mer11, FMP12, FMP13]
which construct a Floer theory for particular classes of magnetic fields.
This thesis would like to give its contribution to the understanding of magnetic
fields on closed orientable surfaces by studying the contact geometry of the level
sets Σm. In reference to point a), b), and c) presented above, this will allow us to
a) count periodic orbits in a given free homotopy class ν using algebraic invariants
such as the Symplectic Cohomology SHν of the sublevels ({E ≤ m22 }, ωσ);
b) find Poincare´ sections starting from pseudoholomorphic foliations [HWZ98,
HLS13];
c) determine whether a periodic orbit is elliptic or hyperbolic by looking at its
ConleyZehnder index [DDE95, AM].
1.2. Contact hypersurfaces and Symplectic Cohomology
We say that a closed hypersurface Σ in a symplectic manifold (W,ω) is of contact
type if there exists a primitive of ω on Σ which is a contact form. Hypersurfaces of
contact type have been intensively studied in relation to the problem of the existence
of closed orbits. Indeed, in this case the Hamiltonian dynamics on Σ is the dynamics
of the Reeb flow associated to the contact form, up to a time reparametrisation.
After some positive results in particular cases [Wei78, Rab78, Rab79], in 1978
Alan Weinstein conjectured that every closed hypersurface of contact type (under
some additional homological condition now thought to be unnecessary) carries a
periodic orbit [Wei79]. The conjecture is still open in its full generality, but for
magnetic systems on orientable surfaces is a consequence of the solution to the
conjecture for every closed 3manifolds by Taubes [Tau07] (see also [Hut10] for an
expository account). Such proof uses Embedded Contact Homology (another kind
of algebraic invariant) to count periodic orbits. Recently, CristofaroGardiner and
Hutchings have refined Taubes’ approach and raised the lower bound on the number
of periodic orbits in dimension 3 to two [CGH12]. The case of irrational ellipsoids
in C2 shows that their estimate is sharp, at least for lens spaces [HT09].
1.2. CONTACT HYPERSURFACES AND SYMPLECTIC COHOMOLOGY 3
When Σ bounds a compact region WΣ ⊂W , we can also compare the orientation
induced by the ambient manifold and the one induced by the contact form. We say
that the hypersurface is of positive contact type if these two orientations agree and
it is of negative contact type otherwise. The importance of such distinction relies
on the fact that in the positive case, and under some additional assumption on the
Chern class of ω, we can define the Symplectic cohomology SH∗ of (WΣ, ω). Its
defining complex is generated by the cohomology of WΣ and the periodic orbits on
Σ. Since the cohomology of the interior is known, if we can compute SH∗, we gain
information about the periodic orbits on the boundary. As is typical in Floer theory,
such computation is divided into two steps:
(1) finding explicitly SH∗ in simple model cases;
(2) proving that SH∗ is invariant under a particular class of deformations, that
bring the case of interest to one of the models.
We prove such invariance in Chapter 3 in a setting that will be useful for the
applications to magnetic flows on surfaces. This result has been inspired to us by
reading [Rit10], where a similar invariance is proven in the setting of ALE spaces
(see [Rit10, Theorem 33 and Lemma 50]. We warmly thank Alexander Ritter for
several useful discussions on this topic. Let us now give the precise statement.
Theorem A. Let W be an open manifold and let ωs be a family of symplectic
forms on W , with s ∈ [0, 1]. Suppose Ws ⊂ W is a family of zerocodimensional
embedded compact submanifold in W , which are all diffeomorphic to a model W ′.
Let (Ws, ωs
∣∣
Ws
, js) be a convex deformation. Fix a free homotopy class of loops ν in
W ′ 'Ws such that the contact forms α0 and α1 are both νnondegenerate. If c1(ωs)
is νatoroidal for every s ∈ [0, 1] and s 7→ ωs is projectively constant on νtori, then
SH∗ν (W0, ω0, j0) ' SH∗ν (W1, ω1, j1).
Before commenting on this result, we clarify the terminology used. We refer
to Chapter 3 for a more thorough discussion. The map js denotes a collar of the
boundary. The normal vector field associated to js induces the contact form αs on
∂Ws. By convex deformation, we mean a deformation for which the boundary stays
of positive contact type. A contact form is νnondegenerate, if all its periodic orbits
in the class ν are transversally nondegenerate. A twoform ρ on W ′ is said to be
νatoroidal if the cohomology class of its transgression [τ(ρ)] ∈ H1(LνW ′) is zero.
A family of twoforms {ω′s} on W ′ is said to be projectively constant on νtori, if the
class [τ(ω′s)] is independent of s up to a positive factor.
Symplectic Cohomology for manifolds with boundary of positive contact type
was introduced by Viterbo in [Vit99]. In [Vit99, Theorem 1.7] (see also [Oan04,
Theorem 2.2]) he proves the invariance of SH∗ for contractible loops when all the ωs
are 0atoroidal (in other words, they are aspherical). However, we do not understand
his proof. In particular, when he deals with the “Generalized maximum principle”
(Lemma 1.8), the coordinate z is treated in the differentiation as if it did not depend
explicitly on the variable s in contrast to what happens in the general case.
4 1. INTRODUCTION
In [BF11], Bae and Frauenfelder prove a similar result for the Symplectic Co
homology of contractible loops on a closed manifolds and for Rabinowitz Floer Ho
mology of twisted cotangent bundles. In their setting all the ωs are aspherical and
the primitives of ωs to the universal cover of W grow at most linearly at infinity.
It would interesting to find out whether the deformation invariance that we get in
Theorem A and the one Bae and Frauenfelder get in [BF11] fit into the long exact
sequence between Symplectic Homology, Symplectic Cohomology and Rabinowitz
Floer Homology [CFO10].
As far as the hypotheses on the symplectic forms are concerned we observe two
things. First, the fact that c1(ωs) is νatoroidal has two consequences: Symplectic
Cohomology is welldefined since c1(ωs) is also aspherical and, hence, no holomor
phic sphere can bubble off [HS95]; Symplectic Cohomology is Zgraded. Second,
the fact that ωs is projectively constant on νtori implies that the local system of
coefficients associated to τ(ωs), which appears as a weight in the definition of the
Floer differential, is independent of s up to isomorphism.
We construct the isomorphism mentioned in Theorem A in two steps. First,
in virtue of Gray’s Theorem we find an auxiliary family of symplectic manifolds
(W ′s, ω′s) such that (W ′s, ω′s) is isomorphic to (Ws, ωs) and the support of ω′s − ω′0
is disjoint from the boundary. Then, we define a class of admissible paths of pairs
{(Hs, Js)} such that Hs is uniformly small and Js is uniformly bounded on the
support of ω′s − ω′0. In this way the 1periodic orbits of XHs does not depend on
s. The moduli spaces of Floer cylinders corresponding to {(Hs, Js)} satisfy uniform
bounds on the energy, since by a PalaisSmale Lemma, the time they spend on the
support of ω′s − ω′0 is uniformly bounded. This implies that the moduli spaces are
compact and, therefore, we can use them to define continuation homomorphisms
between the chain complexes SC∗ν (Ws, ω′s, j′s, Hs, Js). Such homomorphisms will be
weighted using the local system of coefficients associated to τ(ωs), so that they will
commute with the Floer differentials, yielding maps in cohomology. Two homotopies
of homotopies arguments show that these maps are isomorphisms and that they
commute with the direct limit.
In Chapter 4 we apply the general theory of contact hypersurfaces to magnetic
systems (M, g, σ), in order to prove the existence of periodic orbits. We denote by
Con+(g, σ), respectively Con−(g, σ), the set of all m such that Σm is of positive,
respectively negative, contact type.
Historically, the first examples that have been studied are exact systems, since
these can be equivalently described using tools from Lagrangian mechanics. In this
case we can define m0(g, σ) and m(g, σ) the Man˜e´ critical values of the abelian and
universal cover, respectively (after the reparametrisation m 7→ m22 ). As far as the
contact property is concerned, it is known that
• for m > m0(g, σ), m ∈ Con+(g, σ) and, up to time reparametrisation, the
dynamics is given by the geodesic flow of a Finsler metric [CIPP98];
1.2. CONTACT HYPERSURFACES AND SYMPLECTIC COHOMOLOGY 5
• for m ≤ m0(g, σ), Σm is not of restricted contact type, namely the contact
form cannot be extended to the interior, since there exists Mm ⊂ M a
compact manifold with nonempty boundary such that ∂Mm is the union of
supports of periodic orbits whose total action is negative [Ta˘ı91, CMP04].
Each of these orbits is a waist : namely a local minimiser for the action
functional. If M 6= T2, this implies that Con+(g, σ) = (m0(g, σ),+∞),
while, if M = T2, there are examples such that m0(g, σ) ∈ Con+(g, σ)
[CMP04].
As far as the periodic orbits are concerned, it is known that
• for m > m(g, σ) there exists at least a periodic orbit in every nontrivial
free homotopy class;
• for almost every m ≤ m(g, σ) there is a contractible periodic orbit [Con06]
and infinitely many periodic orbits in the same homotopy class of a waist
[AMP13, AMMP14]. Each of these orbits is a mountain pass, namely is
obtained by a minimax argument on a 1dimensional family of loops.
We give a survey of the results about the contact property in Section 4.1. The
only original element we add is an explicit construction of the contact structures at
m0(g, σ) for the twotorus. Indeed, the argument in the original paper [CMP04]
was not constructive. As in the supercritical case, the Symplectic Cohomology of
the filling is isomorphic to the singular homology of the loop space. In particu
lar, SH∗0 6= 0. For restricted contact type hypersurfaces, Ritter proved in [Rit13,
Theorem 13.3] that the nonvanishing of Symplectic Cohomology implies the non
vanishing of Rabinowitz Floer Homology (see [CFO10] for the relation between
these two homology theories). By Theorem 1.2 in [CF09], the nonvanishing of
RFH implies the nondisplaceability of the boundary. Hence, if Ritter’s theorem
could be generalised to the filling of arbitrary hypersurfaces of contact type, we
would have proven that the critical energy level is nondisplaceable (see [CFP10]
for a discussion of displaceability of hypersurfaces in twisted tangent bundles).
We proceed to the nonexact case in Section 4.2. The first condition we need
is for the 2form ωσ to be exact on Σm. This happens if and only if M 6= T2.
Then, we look for a contact form within the class of primitives defined in (4.8). In
Corollary 4.15 and Corollary 4.16, we find that m ∈ Con+(g, σ) for m large enough
and we compute the Symplectic Cohomology of the filling in terms of the Symplectic
Cohomology of the model cases.
Proposition B. Let (M, g, σ) be a magnetic system on a surface different from
the twotorus. If m belongs to the unbounded component of Con+(g, σ), then
SH∗
({
E ≤ m
2
2
}
, ωσ
)
=
H−∗(LM,Z) if M has positive genus,0 if M = S2.
If the magnetic form is symplectic we also show in Section 4.2.2 and Section
4.2.3 that low energy levels are
• of positive contact type, if M = S2;
6 1. INTRODUCTION
• of negative contact type, if M is a surface of higher genus.
In the first case, we compute SH and we again find that it vanishes. In the second
case, we cannot define SH of the filling, since the Liouville vector field points inwards
at the boundary. However, it would be interesting to find a compact symplectic
manifold (W ′, ω′) with boundary of positive contact type that can be glued to Σm
in such a way that ({E ≤ m22 }, ωσ) unionsqΣm (W ′, ω′) is a closed symplectic manifold.
Finally, in Section 4.4, we use the computation of Symplectic Cohomology to
reprove some known lower bounds on the number of periodic orbits in the exact and
nonexact case. We collect them in the following proposition.
Proposition C. Let (M, g, σ) be a magnetic system and suppose that m belongs
either to the unbounded component of Con+(g, σ), or, if M = S2 and σ is symplectic,
to the component of Con+(g, σ) containing 0. Then,
• if M = S2, there exists a periodic orbit on Σm. If all the iterates of this
periodic orbit are nondegenerate, then there exists another geometrically
distinct periodic orbit;
• if M = T2, there exists a periodic orbit on Σm in every nontrivial free
homotopy class. If such orbit is nondegenerate, then there exists another
geometrically distinct periodic orbit in the same class;
• if M is a surface of higher genus, there exists one periodic orbits on Σm in
every nontrivial free homotopy class.
1.3. The contact property on surfaces of revolution
In Chapter 5 we look at the contact property on surfaces of revolution, in order
to test the general results contained in Chapter 4 in a concrete case. We construct
the surface S2γ ⊂ R3 by rotating a profile curve (γ, δ) ⊂ R2 parametrised by arc
length. We study the magnetic system (S2γ , gγ , µγ) where gγ is the restriction of the
Euclidean metric on R3 to S2γ and µγ is the area form associated to this metric. Up
to a homothety, we also assume that the area of the surface is 4pi. From the previous
discussion, we know that there exists two positive values m−,γ < m+,γ such that
[0,m−,γ) ∪ (m+,γ ,+∞) ⊂ Con+(gγ , µγ) =: Conγ . These can be taken to be the two
roots of the quadratic equation m2−mγm+1 = 0 (where mγ ≥ 0 is defined below) if
such roots are real. In this case mγ ≥ 2 and the length of the gap m+, gamma−m−,γ
is
√
m2γ − 4 and, hence, it increases with mγ . If the roots are not real, or, in other
words, mγ < 2, then we simply have Conγ = [0,+∞). The number mγ depends on
the Riemannian geometry of S2γ . It is defined by
mγ := inf
β∈P(1−K)µγ
‖β‖,
where K is the Gaussian curvature and P(1−K)µγ is the set of primitives of (1−K)µγ .
We give an explicit formula for mγ in terms of the function γ in Proposition 5.5. It
can be used to get the following estimate on the contact property.
1.4. LOW ENERGY LEVELS OF CONTACT TYPE ON THE TWOSPHERE 7
Proposition D. If S2γ is symmetric with respect to the equator and the curvature
increases from the poles to the equator, then mγ ≤ 1. Therefore, Conγ = [0,+∞).
On the other hand, for every C > 0, there exists a convex S2γ such that mγ > C.
The second part of the proposition above relies on the fact that mγ can be
arbitrarily big provided the curvature is sufficiently concentrated around at least
one of the poles. Thus, for these surfaces, the contact property can be proved only
for a small set of parameters.
Hence, we are led to ask, in general, how good is the set [0,m−,γ)∪ (m+,γ ,+∞)
in approximating Conγ , the actual set of energies where the contact property holds.
For this purpose we employ McDuff’s criterion [McD87], which says that Σm is
of contact type provided all the invariant measures supported on this hypersurface
have positive action. Finding the actions of an invariant measure is usually a difficult
task. However, for surfaces of revolution there are always some latitudes that are
the supports of periodic orbits. We compute the action of such latitudes in Propo
sition 5.10. If the magnetic curvature Km : Σm → R, defined as Km := m2K + 1 is
positive, we only have two periodic orbits which are latitudes (see Proposition 5.9).
By Proposition 5.10, their action is positive. Therefore, they do not represent an
obstruction to the contact property. In addition, under the same curvature assump
tion, we have a simple description of the dynamics of the system after reduction by
the rotational symmetry. In particular, this allows us to devise a numerical strategy
to compute the action of all the ergodic invariant measures as we explain in Section
5.4. The data we have collected suggest that all such actions are positive, hinting,
therefore, at the following conjecture.
Conjecture E. Let (S2, g, σ) be a symplectic magnetic system and suppose that
for some m > 0, the magnetic curvature is positive. Then, Σm is of contact type.
The numerical computations, and possibly an affirmative answer to the conjec
ture, would then indicate that the system (S2γ , gγ , µγ) associated to a convex sur
face would be of contact type at every energy level. This shows that the inclusion
[0,m−,γ)∪(m+,γ ,+∞) ⊂ Conγ will be strict, in general. Establishing the conjecture
would also yield another proof of Corollary 1.3 in [Sch12a] about the existence of
two closed orbits on every energy level, when K ≥ 0 and f > 0.
To complete the picture, we see in Proposition 5.11 that positive magnetic cur
vature is not necessary for having the contact property. Moreover, using again
Proposition 5.10, we give the first known example of an energy level on a nonexact
magnetic system on S2 which is not of contact type (Proposition 5.12). This shows
that the inclusion Conγ ⊂ [0,+∞) can be strict, as well.
1.4. Low energy levels of contact type on the twosphere
In the last chapter of the thesis we focus on low energy levels of general symplectic
magnetic systems on S2. We can write σ = fµ, where µ is the Riemannian area
and f : S2 → R is a function. Without loss of generality we assume that ∫S2 σ = 4pi
8 1. INTRODUCTION
and that f is positive. By the previous discussion, we know that these levels are of
contact type. Hence, there exists a family of primitives m 7→ λm of ωσΣm made of
contact forms. In Lemma 7.4 we see that, using Gray Stability Theorem, one can
find diffeomorphisms Fm : Σ1 → Σm such that
λˇm := F
∗
mλm =
λˇ0
ρm
+ o(m2), where ρm := 1− m
2
2f
The form λˇ0 is an S
1connection on SS2 := Σ1 with curvature σ. In Lemma 7.14
we show that the Reeb vector field of λˇ0/ρm can be written as the composition of
the σHamiltonian flow with Hamiltonian ρm on the base S
2 and a rotation in the
fibres with angular speed ρm. This allows us to expand Ginzburg’s action function
Sm : SS
2 → R for the form λˇm, defined in Section 7.2, in the parameter m around
zero. This function was introduced for the first time in [Gin87] using local Poincare´
sections. Its critical points are those periodic orbits of the Reeb flow of λˇm which
are close to a curve that winds once around a fibre of SS2 → S2.
Proposition F. The following expansion holds
Sm(x, v) = 2pi +
pi
f(x)
m2 + o(m2). (1.1)
As a consequence, if x ∈ S2 is a nondegenerate critical point of f , then there exists
a family of loops m 7→ γm, such that γ0 winds uniformly once around SxS2 in the
positive sense and the support of Fm(γm) is a periodic orbit on Σm.
The existence of periodic orbits close to nondegenerate critical points is stated
without proof in [Gin96].
Now we move to analyse the ConleyZehnder indices of the Reeb flow of λm and
present some dynamical corollaries of this analysis. As M = S2, Σm is diffeomorphic
to the lens space L(2, 1) ' RP3. On lens spaces we can identify the distinguished
class of dynamically convex contact forms. We say that a contact form is dynamically
convex if the ConleyZehnder index µCZ of every contractible periodic orbit of the
Reeb flow is at least 3. Such forms were introduced by Hofer, Wysocki and Zehnder
[HWZ98] on S3 as a contactinvariant generalisation of convex hypersurfaces in C2.
If τ ∈ Ω1(L(p, q)) is a dynamically convex contact form, we have the following two
implications on the dynamics of the associated Reeb flow Rτ .
(i) There exists a global Poincare´ section of disctype for Rτ , under the condition
that, when p > 1 (namely L(p, q) 6= S3), all periodic orbits are nondegenerate.
(ii) When p > 1 and all periodic orbits are nondegenerate, there exists an elliptic
periodic orbit for Rτ .
Point (i) was proven in [HWZ98] for p = 1. The proof for p > 1 is contained in
[HLS13]. Probably the nondegeneracy assumption can be removed by running the
same lengthy approximation argument contained in [HWZ98]. Point (ii) was proven
for convex hypersurfaces in C2 symmetric with respect to the origin in [DDE95].
A proof of the general case stated above was recently announced in [AM].
1.4. LOW ENERGY LEVELS OF CONTACT TYPE ON THE TWOSPHERE 9
Dynamical convexity was proven in the context of the standard tangent bundle
by Harris and G. Paternain [HP08]. They showed that, if (S2, g) is a Riemannian
twosphere with 1/4pinched curvature, the geodesic flow is dynamically convex on
every energy level. In this thesis we prove dynamical convexity for twisted tangent
bundles (see also the forthcoming paper [Ben14]).
Theorem G. Let (S2, g, σ) be a symplectic magnetic system. If m is low enough,
there exists a primitive λm of ωσ on Σm which is a dynamically convex contact form.
We give two different arguments to show this result.
• In Section 7.3, we construct a hypersurface Σˆm ⊂ C2 and a double cover
pm : Σˆm → Σm such that p∗mλm = −λstΣˆm . The hypersurface Σˆm is convex
for small m, since it tends to the sphere of radius 2 when m goes to zero.
• In Section 7.4, we give a direct estimate of the ConleyZehnder index of
contractible periodic orbits of Rλm .
The latter argument follows closely the strategy of proof of [HP08]: the fact that
the magnetic form is symplectic and the energy is low, plays the same role as the
pinching condition on the curvature. Moreover, this second proof can be adapted to
surfaces of higher genus to show that, if γ is a periodic solution of Rλm which is free
homotopic to eM  times a vertical fibre, then µCZ(γ) ≤ 2eM + 1 (here eM denotes
the Euler characteristic). This inequality fits into a notion of generalised dynamical
convexity which is currently being developed by Abreu and Macarini [AM]. They
claim that they can prove the existence of an elliptic periodic orbit in this wider
setting (see Point (ii)). However, it is not known so far, if it is possible to extend
results on the existence of global Poincare´ sections to this case (see Point (i)).
Theorem G can be used to obtain the following information on the dynamics.
Corollary H. Let (S2, g, σ) be a symplectic magnetic system and let m be low
enough. On Σm there are either two periodic orbits homotopic to a vertical fibre
or infinitely many periodic orbits. If f has three distinct critical points xmin, xMax
and xnondeg such that xmin is an absolute minimiser, xMax is an absolute maximiser
and xnondeg is nondegenerate, then the second alternative holds. Moreover, if Σm
is nondegenerate
• there exists a Poincare´ section of disctype for the magnetic flow on Σm;
• there exists an elliptic periodic orbit γ on Σm and, therefore, generically,
there exists a flowinvariant fundamental system of neighbourhoods for γ.
Hence, the dynamical system is not ergodic with respect to the Liouville
measure on Σm.
We do not have any example where there are exactly two noncontractible pe
riodic orbits on low energy levels. However, by [Gin87, Assertion 3] we know that
these two orbits are short, namely they are close to curves that wind around a verti
cal fibre once. In the next theorem, we show that we can make rigorous a dichotomy
between short and long orbits.
10 1. INTRODUCTION
Theorem I. Let (S2, g, σ = fµ) be a symplectic magnetic system. Given ε > 0
and a positive integer n, there exists mε,n > 0 such that for every m < mε,n the
projection pi(γ) of a periodic prime solution γ on Σm either is a simple curve on S
2
with length in
(
2pi−ε
max fm,
2pi+ε
min fm
)
or has at least n selfintersections and length larger
than mε .
This result is an adaptation to the magnetic settings of [HS12, Theorem 1.6]
for Reeb flows of convex hypersurfaces close to S3 ⊂ C2. That paper brings in
the contact category classical results for pinched Riemannian metrics on S2 [Bal83]
and on spheres of any dimension [Ban86]. Theorem 1.6 in [HS12] also contain a
lower bound on the linking number between short and long orbits. This statement
is substituted in Theorem I by a lower bound on the number of selfintersections
for long orbits. It is likely that the estimates on the linking number obtained by
Hryniewicz and Saloma˜o can be used as a black box to get our estimates for the
selfintersections, but we did not pursue this strategy explicitly. Instead, our proof
is based on an application of the GaussBonnet formula for surfaces.
Even if we do not know if it is possible to have an energy level with exactly two
periodic orbits, it has been proven by Schneider in [Sch11, Theorem 1.3] that there
are examples with exactly two short orbits. In Section 7.6, we have a closer look
at this problem for surfaces of revolution (S2γ , gγ , fµγ), where f is a rotationally
invariant function. We show the existence of a smooth family of embedded tori
Cm : [−`,+`]/∼ × T2pi ↪→ Σm, where ` is the length of γ and ∼ is the equivalence
relation which identifies −` and +`. They are defined by the formula
Cm(u, ψ) :=
(−u,−pi/2, ψ) if u < 0,(u, pi/2, ψ + pi) if u > 0,
where we have put on Σm the triple of coordinates (t, ϕ, θ). The first and third
one are the latitude and longitude, respectively. The second one is the angle in the
vertical fibre counted starting from the longitudinal direction. Each Cm is obtained
by gluing two global Poincare´ sections of cylindertype, C−m and C+m, along their
common boundary. This boundary is made by the two unique periodic orbits that
project to latitudes on S2γ . The two smooth return maps defined on the interior of
C−m and C+m extend to a global continuous map F 2m : Cm → Cm (Proposition 7.35).
Denote by Ωf : S
2
γ → R the extension of the function − f˙γf3 to the poles and set
Ω−f := inf Ωf . We have the following result for the return map.
Proposition J. The family of maps F 2m : [−`,+`]/∼×T2pi → [−`,+`]/∼×T2pi
admits the expansion
F 2m(u, ψ) =
(
u, ψ + piΩfm
2 + o(m2)
)
.
Hence, if Ωf is not constant, there are infinitely many periodic orbits on every low
energy level. Such condition is satisfied if, for example, f¨
f3
(0) 6= − f¨
f3
(`). If Ω−f > 0
(namely f˙ = 0 only at the poles and f¨ 6= 0 there), the period T of a Reeb orbit of
1.4. LOW ENERGY LEVELS OF CONTACT TYPE ON THE TWOSPHERE 11
Rλm, different from a latitude, satisfies
T ≥ 2
Ω−f m2
+O
(
1
m3
)
.
In particular, there are only two short orbits and their projection to S2γ is supported
on two latitudes.
Finally, we observe that this proposition also gives potential candidates for mag
netic systems with only two closed orbits. Namely, those for which Ωf is constant.
CHAPTER 2
Preliminaries
In this chapter we set the notation and recall the prerequisites needed in the
subsequent discussion. The first section describes the conventions and symbols used
in the paper. The second section is devoted to the basic properties of the tangent
bundle of an oriented Riemannian surface. The third section introduces magnetic
fields. The fourth section deals with Hamiltonian structures on threemanifolds.
Magnetic flows on a positive energy level are particular instances of this general
class of dynamical systems.
2.1. General notation
All objects are supposed to be smooth unless otherwise specified.
If T > 0 is a real number, we set TT := R/TZ.
If p, q are coprime positive integers, we write L(p, q) for the associated lens space.
It is the quotient of S3 ⊂ C2 by the Z/pZaction given by
[k] · (z1, z2) = (e2pii
k
p z1, e
2pii kq
p z2).
If γ and γ′ are two knots in S3 we denote by lk(γ, γ′) their linking number.
If M is a manifold we write Ωk(M) for the space of kdifferential forms on M and
Γ(M) for the space of vector fields on M . The interior product between Z ∈ Γ(M)
and ω ∈ Ωk(M) will be written as ıZω. If ω ∈ Ωk(M), we denote by Pω the set of
its primitives. Namely, Pω = {τ ∈ Ωk−1(M)  ω = dτ}.
If Z ∈ Γ(M), we denote by LZ the associated Lie derivative and by ΦZ the flow
of Z defined on some subset of R ×M . We write its time t flow map as ΦZt . A
periodic orbit for Z is a loop γ : TT → M , such that γ˙ = Zγ . When we want to
make the period of the orbit explicit we use the notation (γ, T ). We call ΩkZ(M) the
space of Zinvariant kforms: τ belongs to ΩkZ(M) if and only if LZτ = 0.
If ν is a free homotopy class of loops in M , we denote by LνM the space of
loops in ν.
Let pi : E → M be an S1bundle over a surface M with orientation oM . We
denote by V the generator of the S1action and we endow E with the orientation
OE := oM ⊕−V . An S1connection form on E is a τ ∈ Ω1(E) such that τ(V ) = 1
and dτ = −pi∗σ for some σ ∈ Ω2(M) called the curvature form.
If (M,ω) is a symplectic manifold and H : M → R is a real function, the
Hamiltonian vector field XH is defined by
ıXHω = −dH. (2.1)
13
14 2. PRELIMINARIES
We define now some objects on Cn ' R2n. Denote by Jst the standard complex
structure and by gst the Euclidean inner product. Define the standard Liouville form
λst ∈ Ω1(Cn) as (λst)z(W ) := 12gst(Jst(z),W ), for z ∈ Cn and W ∈ TzCn ' Cn.
Finally, the standard symplectic form is defined as ωst := dλst, or in standard real
coordinates ωst =
∑
i dx
i ∧ dyi.
2.2. The geometry of an oriented Riemannian surface
Let M be a closed orientable surface and let eM ∈ Z be its Euler characteristic.
Let pi : TM → M be the tangent bundle of M , LV(x,v) : TxM → T(x,v)TM the
associated vertical lift and write TM0 for the complement of the zero section in
TM . Suppose that we also have fixed an orientation o on M . If σ ∈ Ω2(M) we use
the shorthand [σ] :=
∫
M σ (where the integral is with respect to o).
Let g be a Riemannian metric on M . It yields an isomorphism [ : TM → T ∗M
which we use to push forward the metric for tangent vectors to a dual metric g for
1forms. Call ] : T ∗M → TM the inverse isomorphism. We write  ·  for the induced
norms on each TxM and T
∗
xM . From the duality construction we have the identity
l = supv=1 l(v), ∀ l ∈ T ∗xM . We collect this family of norms together to get a
supremum norm ‖ · ‖ for sections:
∀Z ∈ Γ(M), ‖Z‖ := sup
x∈M
Zx; ∀β ∈ Ω1(M), ‖β‖ := sup
x∈M
βx. (2.2)
The Riemannian metric induces a kinetic energy function E : TM → R defined by
E
(
(x, v)
)
:= 12gx(v, v). The level sets Σm := {E = 12m2} ⊂ TM are such that
• the zero level Σ0 is the zero section {(x, 0) x ∈M};
• for m > 0, pi : Σm →M is an S1bundle.
Let ∇ be the Levi Civita connection of g and ∇γdt be the associated covariant deriv
ative along a curve γ on M . For every (x, v) ∈ TM , ∇ gives rise to a horizontal lift
LH(x,v) : TxM → T(x,v)TM . It has the property that d(x,v)pi ◦LH(x,v) = IdTxM . Finally,
denote by µ ∈ Ω2(M) the positive Riemannian area form and by K the Gaussian
curvature. We combine them to obtain the curvature form σg ∈ Ω2(M) associated
to g. It is defined by σg := Kµ.
We introduce a frame of T (TM0) and a coframe of T
∗(TM0) depending on (g, o).
The first element of the frame is Y , which is defined as Y(x,v) := L
V
(x,v)(v).
The geodesic equation
∇γ
dt
γ˙ = 0 (2.3)
for curves γ in M gives rise to a flow on TM . The second element of the frame
is the generator X of such flow. It is called the geodesic vector field and can be
equivalently defined as X(x,v) = L
H
(x,v)(v).
Consider the 2piperiodic flow ΦVϕ : TM → TM , which rotates every fibre of pi
by an angle ϕ. We choose as the third element of our frame, V the generator of
this flow. If we denote by : TM → TM the rotation of pi/2 in every fibre, then
V(x,v) = L
V
(x,v)(xv) and V
∣∣
Σm
∈ Γ(Σm) is the generator of the S1action on Σm.
2.2. THE GEOMETRY OF AN ORIENTED RIEMANNIAN SURFACE 15
Finally, the last element of the frame is H, defined by H(x,v) := L
H
(x,v)(xv).
The first element of the coframe is dE, the differential of the kinetic energy.
The second element is λ, the pullback under [ of the standard Liouville form
on T ∗M . It acts by λ(x,v)(ξ) := gx(v, d(x,v)pi ξ) on ξ ∈ T(x,v)TM . The exterior
differential ω := dλ yields a symplectic form on TM . We denote by O the orientation
associated to ω ∧ ω. Every Σm inherits an orientation OΣm which is obtained from
O following the convention of putting the outward normal to Σm (namely Y ) first.
Observe that such orientation matches with the orientation of an S1bundle over an
oriented surface as defined in Section 2.1.
The third element is τ , the angular component of the LeviCivita connection.
On ξ ∈ T(x,v)TM , it acts by τ(x,v)(ξ) = gx(xv, xˆdt vˆ), where (xˆ, vˆ) is any path passing
through (x, v) with tangent vector ξ.
The fourth element is given by η := ∗λ.
We now recall the main properties of the frame and the coframe, which are a
particular case of the results contained in [GK02b].
Proposition 2.1. The frame (X,Y,H, V ) is Opositive with dual coframe(
λ
2E
,
dE
2E
,
η
2E
,
τ
2E
)
.
We have the following bracket relations for the frame [Y,X] = X [Y,H] = H [Y, V ] = 0[V,X] = H [V,H] = −X [X,H] = 2EKV. (2.4)
Accordingly, we get the following structural equations for the coframe
d
( λ
2E
)
=
λ
2E
∧ dE
2E
− η
2E
∧ τ
2E
,
d
(dE
2E
)
= 0,
d
( η
2E
)
=
λ
2E
∧ τ
2E
− dE
2E
∧ τ
2E
,
d
( τ
2E
)
= −2EK λ
2E
∧ η
2E
.
(2.5)
The last equation in (2.5) can be rewritten as
d
( τ
2E
)
= −pi∗(Kµ). (2.6)
It implies that τ2E is an S
1connection form on every Σm with curvature σg.
For the following discussion it will be convenient to have also a corresponding
statement for the restriction to SM := Σ1 of the frame and coframe defined above.
Corollary 2.2. The triple (X,V,H) is an OSM positive frame of T (SM) with
dual coframe (λ, τ, η). We have the following relations{
[V,X] = H [H,V ] = X [X,H] = KV ;
dλ = τ ∧ η dτ = Kη ∧ λ = −pi∗σg dη = λ ∧ τ.
(2.7)
16 2. PRELIMINARIES
Definition 2.3. The volume form χ ∈ Ω3(SM) defined by
χ := λ ∧ τ ∧ η = −τ ∧ pi∗µ, (2.8)
associated to the dual coframe, is called the Liouville volume form.
We end up the general discussion about the geometry of M by proving some
properties related to local sections of SM .
Consider an open set U ⊂M and denote by SU the unit sphere bundle over U .
Let Z : U → SU be a section and associate to Z an angular function ϕZ : SU → T2pi.
The value of ϕZ at the point (x, v) is the angle between v and Zx. In the next lemma
we compute the differential of the angular function. First, observe that, if v ∈ TxU ,
we have gx
(
(∇vZ)x, Zx
)
= 0. Thus, there exists κZx ∈ T ∗xM such that
(∇vZ)x = κZx (v)xZx, (2.9)
Lemma 2.4. If Z is a section of SU , we have
dϕZ(X)(x, v) = −κZx (v),
dϕZ(V )(x, v) = 1,
dϕZ(H)(x, v) = −κZx (xv).
(2.10)
Proof. We can write the angle ϕZ as an element of S
1 ⊂ C:
(cosϕZ(x, v), sinϕZ(x, v)) =
(
gx(v, Zx), gx(v, xZx)
)
. (2.11)
Differentiating along X this map, we get(
d(x,v)(cosϕZ)(X), d(x,v)(sinϕZ)(X)
)
=
(
gx((∇vZ)x , v),−gx((∇vZ)x , xv)
)
.
On the other hand, we also have(
d(x,v)(cosϕZ)(X), d(x,v)(sinϕZ)(X)
)
= dϕZ(X)(x, v)·
(−sinϕZ(x, v), cosϕZ(x, v)).
Thus, we can express dϕZ(X)(x, v) as the standard inner product in R2 between(
gx((∇vZ)x , v),−gx((∇vZ)x , v)
)
and
(− sinϕZ , cosϕZ).
dϕZ(X)(x, v) = − sinϕZ(x, v)gx
(
(∇vZ)x , v
)
+ cosϕZ(x, v)gx
(
(∇vZ)x , xv
)
= − sinϕZ(x, v)κZx (v)gx(xZ, v)− cosϕZ(x, v)κZx (v)gx
(
xZ, xv
)
= −κZx (v).
For the second statement we differentiate in t the identity ϕZ ◦ ΦVt = ϕZ + t.
The third statement follows by using the identity H = d
−1(X):
d(x,v)ϕZ(H(x,v)) = d(x,v)ϕZ
(
d(x,xv)
−1(X(x,xv)))
= d(x,xv)(ϕZ ◦ −1)
(
X(x,xv)
)
= d(x,xv)ϕZ
(
X(x,xv)
)
= −κZx (xv)
2.3. MAGNETIC FIELDS 17
We give an explicit formula for κZx in the next lemma. First, define the geodesic
curvature of Z as the function
kZ : U −→ R
x 7−→ gx
(
(∇ZZ)x, xZx
)
.
(2.12)
Lemma 2.5. At every x ∈M , we have ] κZx = kZ(x)Zx + kZ(x)xZx.
Proof. Let us evaluate κZx on the basis (Zx, xZx):
κZx (Zx) = gx
(
(∇ZZ)x, xZx
)
= kZ(x);
κZx (xZx) = gx
(
(∇ZZ)x, xZx
)
= gx
(
x(∇ZZ)x, x(xZx)
)
= gx
(
(∇ZZ)x, x(xZx)
)
= kZ(x).
2.3. Magnetic fields
In the previous section we introduced various geometrical objects associated to
(M, g). Historically, the geodesics equation (2.3) has been used as the main tool to
understand the geometry of Riemannian manifolds [Kli95, Pat99a]. It is a classical
result that the geodesic flow can be studied in the framework of symplectic geometry,
since X is the dλHamiltonian vector field associated to the kinetic energy function
(see [Kli95, Chapter 3]). In other words,
ıXdλ = −dE. (2.13)
From this equation we see that Σ0 is the set of rest points for X and for every
positive m, Σm is invariant under the geodesic flow. Hence, we can study X
∣∣
Σm
for a fixed value of m > 0. By the homogeneity of (2.3) (which is translated in
the bracket relation [Y,X] = X), X
∣∣
SM
and X
∣∣
Σm
have the same dynamics up to
reparametrisation by a constant factor. The map conjugating the two flows is given
by the flow of Y :
dΦY− logm ·X
∣∣
Σm
= mX
∣∣
ΣSM
(2.14)
One of the guiding principles of this thesis is to see how the introduction of a
magnetic perturbation in the geodesic equation breaks the homogeneity and gives
rise to a family of systems whose dynamics changes with m.
For any σ ∈ Ω2(M) consider the symplectic form ωσ := dλ − pi∗σ ∈ Ω2(TM).
When σ is not zero, we refer to the symplectic manifold (TM,ωσ) as a twisted
tangent bundle to distinguish it from the standard tangent bundle (TM,ω0 = dλ).
We will see later some remarkable differences (in terms of Symplectic Coho
mology and displaceability, for example) between the geometry of the standard and
twisted tangent bundle. Such phenomena show, in this setting, the contrast between
symplectic and volumepreserving geometry. Indeed, we claim that
ωσ ∧ ωσ = ω0 ∧ ω0. (2.15)
18 2. PRELIMINARIES
Hence, the twisted and the standard tangent bundle have the same volume form,
although they are different in the symplectic category. To prove the claim notice
that pi∗σ ∧ pi∗σ = 0 and that dλ ∧ pi∗σ = d(λ ∧ pi∗σ) = 0, since λ ∧ pi∗σ = 0 by
Proposition 2.1 above.
In the next proposition we prove that also the first Chern class of the underlying
class of compatible almost complex structures is unaffected by the twist.
Proposition 2.6. If σ is a 2form on M , then c1(TM,ωσ) = 0.
Proof. Consider the vertical distribution V → TM . It is a Lagrangian sub
bundle of the symplectic bundle (T (TM), ωσ) → TM . If Jσ is an almost complex
structure compatible with ωσ, then V is totally real with respect to Jσ. As a conse
quence, we have the isomorphism of complex bundles
(VC, i) −→ (T (TM), Jσ)
(u+ iv) 7−→ u+ Jσv,
where (VC, i) is the complexification of V. Therefore, c1(TM, Jσ) = c1(VC, i). On
the other hand, c1(VC, i) = 0 since (VC, i) ' (VC,−i), c1(VC,−i) = −c1(VC, i) and
H2(TM,Z) is torsionfree (see [MS74, Lemma 14.9]. Alternatively, notice that
V → TM is orientable and therefore det(V) → TM is trivial. Since taking the
determinant bundle and taking the complexification do commute, we get
c1(VC, i) = c1(det(VC), det(i)) = c1((detV)C, i) = 0.
Definition 2.7. We call σ a magnetic form and the triple (M, g, σ) a magnetic
system. If we fix M , we write Mag(M) for the space of all pairs (g, σ) such that
(M, g, σ) is a magnetic system. For every (g, σ) ∈ Mag(M), there exists a unique
function f : M → R called the magnetic strength such that σ = fµ. We say that the
magnetic system is exact, nonexact, or symplectic, if such is the magnetic form σ in
Ω2(M). We denote by Mage(M) ⊂ Mag(M) the subset of exact magnetic systems.
Finally, we define the magnetic vector field XσE ∈ Γ(TM) as the ωσHamiltonian
vector field associated to E and we refer to ΦX
σ
E as the magnetic flow.
As the geodesic flow, XσE comes from a second order ODE for curves in M :
∇γ
dt
γ˙ = Fγ
(
γ˙
)
. (2.16)
Here F : TM → TM is the Lorentz force. It is a bundle map given by
g(Fx(v), w) := σx(v, w) (2.17)
and can be expressed using the magnetic strength as Fx(v) = f(x)xv. Using the
relation between the Levi Civita connection and the horizontal lifts, one finds that
XσE = X + fV. (2.18)
The kinetic energy E is still an integral of motion for the magnetic flow: Σ0 is
the set of rest points for XσE and, for m > 0, X
σ
E restricts to a nowhere vanishing
vector field on Σm.
2.3. MAGNETIC FIELDS 19
However, the bracket relations now imply
dΦY− logm ·XσE
∣∣
Σm
= (mX + fV )
∣∣
SM
= mX
σ
m
E
∣∣
SM
. (2.19)
From this equality we infer two things. First, that, unlike the geodesic flow, the
magnetic flow is not homogeneous. Second, that studying the dynamics of the
magnetic system (M, g, σ) on Σm is the same as studying the rescaled magnetic
system (M, g, σm) on SM . The advantage of the latter formulation is that it describes
the dynamics of the magnetic flow by using a 1parameter family of flows on a fixed
closed threemanifold. We also introduce the real parameter s, which is related to
m by the equation sm = 1. In the following discussion, we will use both parameters,
choosing every time the one that most simplify the notation. As a heuristic rule,
the parameter m will be more convenient for low values of E (m close to 0), while
s will be more convenient for high values of E (s close to 0). When there is no risk
of confusion, we adopt the shorthand Xs := XsσE
∣∣
SM
and ωs := ωsσ, when we use s,
and Xm := mX
σ
m
E
∣∣
SM
= mX + fV and ωm := mω σ
m
= mdλ− pi∗σ, when we use m.
Notice that we have the limits
lim
s→0
Xs = X, lim
m→0
Xm = fV,
lim
s→0
ωs = dλ; lim
m→0
ωm = −pi∗σ.
In the standard tangent bundle, the properties of the linearisation of the geodesic
flow on SM are influenced by the curvature K (see also the bracket relations above).
Analogously, when there is a magnetic term, for every m > 0, we can define the
magnetic curvature function Km : SM → R as Km := m2K −m(df ◦ ) + f , which
is related to the linearisation of Xm [Ada97] (see also Chapter 5).
Remark 2.8. The duality isomorphism [ : TM → T ∗M yields a symplectomor
phism between the twisted tangent bundle (TM,ωσ) and (T
∗M,ω∗σ := dλ∗ − pi∗σ),
where λ∗ is the standard Liouville form on the cotangent bundle and we denote
with pi also the footpoint projection T ∗M → M . The ωσHamiltonian system XσE
is sent through [ to the ω∗σHamiltonian system associated to E∗ := E ◦ [−1. Ob
serve that E∗ is the kinetic energy function on T ∗M associated to the dual metric,
namely E∗(x, l) = 12gx(l, l). In particular Σ
∗
m := [(Σm) = {E∗ = 12m2}. Unlike ωσ,
ω∗σ depends only on σ and not on g, which enters in the picture only through the
Hamiltonian E∗.
The tangent bundle and the cotangent bundle formulation are equivalent but
have different advantages and disadvantages as far as the exposition of the material
is concerned. In the tangent bundle we have an intuitive understanding of the flow
given by the second order ODE. On the other hand, in the cotangent bundle we
have seen that g and σ contributes separately to determine the magnetic flow: the
former through the Hamiltonian and the latter through the symplectic form.
We have seen that looking at different energy levels of a fixed magnetic system
(M, g, σ) corresponds to a suitable rescaling of the magnetic field: XσE
∣∣
Σm
∼ X
σ
m
E
∣∣
Σ1
.
20 2. PRELIMINARIES
What happens when we rescale the metric g 7→ m2g? In the cotangent bundle for
mulation, we have that E∗m2g =
E∗g
m2
. Thus, XσE∗
m2g
∣∣
Σ∗1
∼ XσE∗g
∣∣
Σ∗m
and, consequently,
XσEm2g
∣∣
Σ1
∼ XσEg
∣∣
Σm
. (2.20)
2.3.1. The Maupertuis’ principle. In this subsection we discuss how a mag
netic system (M, g, σ) is affected by the introduction of a potential U : M → R. In
this case we speak of a mechanical system (M, g, σ, U). The Hamiltonian has an
additional term E(x, v) = 12gx(v, v) + U(x) while the symplectic form remains the
same ωσ = dλ − pi∗σ. In this case the topology of the energy levels depends on
U . Consider the map pi : Σm → M , where Σm := {E = m2/2}. Its image is the
sublevel Mm := {U ≤ m2/2} and the preimage of a point x ∈Mm are the vectors in
TxM with norm
√
m2 − 2U(x). This is either a single point, namely the zero vector
of TxM , if U(x) = m
2/2, or a circle, if U(x) < m2/2. In particular, pi : Σm →M is
an S1bundle if and only if m2/2 > maxU . We claim that in this case the dynamics
on Σm is equivalent, up to reparametrisation, to the dynamics of the magnetic flow
associated to (M, gm,U := (m
2 − 2U)g, sigma) on SM (where the sphere bundle is
taken with respect to the metric gm,U ). To prove the claim, we work in the cotangent
bundle formulation. We start by observing that {E∗ = m2/2} = {12 gx(l,l)m2−2U(x) = 12}.
Therefore, the ω∗σHamiltonian flow of E∗ and the one of E˜∗(x, l) :=
1
2
gx(l,l)
m2−2U(x) are
the same up to reparametrisation on this common energy level. Finally, we observe
that E˜ is the kinetic energy function for the metric g
m2−2U which is the dual of gm,U .
From the discussion above, we see that the results contained in this thesis can
also be applied to magnetic systems with potential, provided we know U and m well
enough to understand the properties of (M, gm,U , σ). For example, fix m > 0 and
take a potential U whose C0norm is small compared to m2. We claim that the
mechanical system (M, g, σ, U) and the magnetic system (M, g, σ) will be close to
each other on Σm. By Equation (2.19) and (2.20), we know that the magnetic system
associated to (M, gm,U , σ) is equivalent, up to reparametrisation, to the magnetic
system associated to (M,
gm,U
m2
, σm). Moreover,∥∥∥gm,U
m2
− g
∥∥∥ ≤ 2‖U‖
m2
‖g‖. (2.21)
Thus, if ‖U‖ is small compared to m2, the magnetic flow of (M, gm,U , σ) on the unit
tangent bundle of gm,U is close, up to reparametrisation, to the magnetic flow of
(M, g, σm) on the unit tangent bundle of g.
2.3.2. Physical interpretation: constrained particles and rigid bodies
with symmetry. The adjective ’magnetic’ for the systems we have introduced
in the present section is due to the fact that they describe the following simple
phenomenon in the theory of classical electromagnetism. Consider a particle q of
unit mass constrained to move on a frictionless surface M in the Euclidean space
(R3, gst). The metric induces by pullback a metric g on M and a corresponding
LeviCivita connection ∇. With this assumption the motion of q will obey to the
geodesic equation (2.3). Suppose now that q has unit charge and that M is immersed
2.3. MAGNETIC FIELDS 21
in a stationary magnetic field
−→
B . In this case the particle is subject to a Lorentz
force
−→
F = −→v × −→B . The force −→F is obtained from the 2form σ := ı−→
B
volgst as
prescribed by Equation (2.17). Here volgst denotes the Euclidean volume form in
R3. The motion of q will obey to the Equation (2.16).
Making the natural assumption that
−→
B is defined on the whole R3, we see that
σ is the pullback of a closed 2form on R3 and, hence, it is exact. With this physical
interpretation, nonexact magnetic systems on M correspond to fields
−→
B generated
by a Dirac monopole located in a region outside the surface.
Even if a magnetic monopole has not been observed so far, nonexact mag
netic flows on S2 have, nonetheless, a concrete importance since they are symplectic
reductions of certain physical systems with phase space TSO(3), possessing an S1
symmetry. The study of the periodic orbits of these systems was initiated in the
early Eighties by Novikov in a series of papers [NS81, Nov81a, Nov81b, Nov82].
His interest was motivated by the fact that in these cases the Lagrangian action func
tional is multivalued and, hence, one needs a generalisation of the standard Morse
theory (nowadays known as Novikov theory) for proving the existence of critical
points.
Among the many examples considered by Novikov, here we only look at the
motion of a rigid body with rotational symmetry and a fixed point. We refer the
reader to [Kha79] for a careful explanation of the setting and for the proofs of the
statement we make.
A rigid body with a fixed point is described by a positive orthonormal basis
e = (e1, e2, e3) in R3. Given a fixed positive orthonormal basis n = (n1, n2, n3), to
every e we can associate a unique element in SO(3) which is the isometry of R3
sending n to e. Hence, the configuration space of a rigid body with a fixed point
can be identified with the group SO(3).
The kinetic energy of the body is obtained from a leftinvariant Riemannian
metric g on SO(3), which, in the standard basis of so(3), is represented by the
matrix
I =
I1 0 00 I2 0
0 0 I3
(2.22)
for some positive numbers I1, I2 and I3. Let us consider the T2piaction on the
configuration space that rotates the rigid body around the fixed axis n3. In SO(3)
this corresponds to left multiplication by the subgroup G of rotations with axis n3.
Call Z ∈ Γ(SO(3)) the infinitesimal generator of G.
The quotient map for this action is the S1bundle map
p : SO(3) −→ S2 ⊂ R3
(e1, e2, e3) 7−→
α1α2
α3
:=
gst(e1, n3)gst(e2, n3)
gst(e3, n3)
,
22 2. PRELIMINARIES
which yields the coordinates of the vector n3 in the frame n. Denote by gˆ the
metric induced on S2 by p. It is obtained pushing forward the restriction of g to the
orthogonal of ker dp. Notice that the function e 7→√ge(Z,Z) is invariant under G,
so that it descends to a function ν : S2 → (0,+∞) such that √ge(Z,Z) = ν(p(e)).
Finally, suppose that the rigid body is immersed in a conservative force field,
which is invariant under rotations around n3. Thus, the force is described by a
potential U : SO(3) → R which is invariant under the T2piaction. Namely, there
exists Uˆ : S2 → R such that U = Uˆ ◦ p.
The dynamics of the body is given by the Hamiltonian vector field on TSO(3)
associated to the mechanical system (SO(3), g, 0, U). The T2pisymmetry arises from
a momentum map J : TSO(3) → g ' R, which is an integral of motion. For every
k ∈ R, we look at the dynamics on the invariant set {J = k} projected to TS2 via
the quotient map p. It is the dynamics of the Hamiltonian vector field associated
to (S2, gˆ, kσgˆ, Uˆk), where Uˆk = Uˆ +
k2
2ν2
. A computation shows that the Gaussian
curvature of gˆ is always positive and, hence, the magnetic form is symplectic.
Applying Maupertuis’ principle, we know that, on Σm such that m
2/2 > max Uˆk,
the reduced system above is equivalent to the magnetic system (S2, gˆm,Uˆk , kσgˆ).
Moreover, if ‖Uˆk‖ is small compared to m2, up to reparametrisation, the dynamics
is close to that of (S2, gˆ, kmσgˆ) on the unit tangent bundle of gˆ.
2.3.3. From dynamics to geometry. We come back to the general discussion
on magnetic flows associated to (M, g, σ). We saw that the dynamics of XσE on TM
can be studied by looking at the dynamics of Xs := XsσE
∣∣
SM
, for positive values
of the parameter s. We are going to look at this new problem from a geometric
point of view, by investigating what geometric structure the symplectic manifold
TM induces on the threemanifold SM .
Consider the restriction ω′σ := ωσ
∣∣
SM
. It is a closed, nowhere vanishing 2form
on the unit tangent bundle. Hence, it has an orientable 1dimensional kernel kerω′σ
at every point. We fix for it an orientation Okerω′σ satisfying the relation
OSM = Okerω′σ ⊕Oω
′
σ , (2.23)
where Oω
′
σ is the orientation on T (SM)kerω′σ
induced by ω′σ. Since χ = λ ∧ dλ = λ ∧ ω′σ,
we readily see that
ı
XσE
∣∣
SM
χ = λ(XσE
∣∣
SM
)ω′σ − λ ∧ ıXσE
∣∣
SM
ω′σ = ω
′
σ. (2.24)
Therefore, Equation (2.23) implies that the magnetic vector field XσE
∣∣
SM
is a positive
nowhere vanishing section of kerω′σ.
From the previous discussion, we argue that the real object of interest for un
derstanding the geometric properties of the orbits, and not their actual parametri
sation, is kerω′σ rather than XσE
∣∣
SM
. The 2form ω′σ is a particular instance of what
is called a Hamiltonian Structure (or HS for brevity). We introduce such objects in
an abstract setting in the next section.
2.4. HAMILTONIAN STRUCTURES 23
2.4. Hamiltonian structures
We develop below the notion of HS and we identify the special subclass of HS
of contact type. Proving that a HS is of contact type has striking consequences for
the dynamics, since in this case we can describe the dynamical system associated to
the HS by the means of a Reeb flow. This yields the following two advantages.
a) We can count periodic orbits of Reeb flows using algebraic invariants, such as
Contact Homology [Bou09] and Embedded Contact Homology [Tau07], since
closed orbits are generator for the corresponding chain complexes. When the
contact manifold is the positive boundary of a symplectic manifold, like in the
case of SM , one can define a further invariant called Symplectic Homology, which
takes into account the interplay between the contact structure on the boundary
and the symplectic structure of the filling. In the next chapter we are going to
define Symplectic Homology [Vit99, Sei08] (or, more precisely, its cohomological
version) in detail and prove an abstract invariance result which we will use in
Chapter 4 to compute it in the case of magnetic systems. This will yield a lower
bound on the number of periodic orbits for such systems.
b) The theory of pseudoholomorphic foliations developed by Hofer, Wysocki and
Zehnder [HWZ98] allows to find Poincare´ sections for dynamically convex Reeb
flows. This allows for a description of the flow in terms of an areapreserving dis
crete dynamical system on the twodimensional disc. Information about periodic
points of such systems are then obtained applying results by Brouwer [Bro12]
and Franks [Fra92]. The abstract setting will be presented in Chapter 6 and
applied to magnetic systems in Chapter 7
In the rest of this section let (N,O) be a threemanifold N endowed with an
orientation O.
Definition 2.9. A closed and nowhere vanishing 2form ω on N is called a
Hamiltonian Structure. We say that the Hamiltonian Structure is exact, if ω is
such. The 1dimensional distribution kerω is called the characteristic distribution
associated to ω. It has an orientation Okerω satisfying the relation O = Okerω⊕Oω,
where Oω is the orientation on TNkerω induced by ω. Alternatively, if χ is any positive
volume form, Zχ ∈ Γ(N) defined by ıZχχ = ω is a positive section of kerω.
Definition 2.10. Let ω be a HS and let Z ∈ Γ(N) be a positive section of the
characteristic line bundle associated with ω. We say that a periodic orbit (γ, T )
for ΦZ is nondegenerate if the multiplicity of 1 in the spectrum of dΦZT is exactly
one. Observe that this property does not depend on the choice of Z. If (γ, T )
is transversally nondegenerate, call transverse spectrum the set made of the two
eigenvalues of dΦZT different from 1. We say that γ is
(1) elliptic, if the transverse spectrum lies on the unit circle in C;
(2) hyperbolic, if the transverse spectrum lies on the real line.
Finally, we say that ω is νnondegenerate if all the periodic orbits in the class
ν are nondegenerate.
24 2. PRELIMINARIES
Remark 2.11. If (W,ω) is a symplectic fourmanifold and N ↪→ W is an ori
entable embedded hypersurface, then ω
∣∣
N
is a Hamiltonian Structure on W . If we
take N to be the regular level of a Hamiltonian function H : W → R, we can endow
N with the orientation induced from ω ∧ ω by putting the gradient of H first. Us
ing the Hamilton equation (2.1), we see that XH
∣∣
N
is a positive section of kerω
∣∣
N
.
Moreover, ω
∣∣
N
is νnondegenerate if and only if all the periodic orbits of H on N
with free homotopy class ν are transversally nondegenerate (see next chapter).
This observation shows that the dynamics up to reparametrisation of an au
tonomous Hamiltonian system on a fourdimensional symplectic manifold (W,ω)
at a regular level N can be read off the geometry of the oriented onedimensional
distribution kerω
∣∣
N
associated to the Hamiltonian structure ω
∣∣
N
.
2.4.1. Hamiltonian structures of contact type. We now introduce Hamil
tonian structures of contact type for which the characteristic distribution has a
section with special properties.
Definition 2.12. We say that ω is of contact type if it is exact and there exists a
contact form τ ∈ Pω. Denote by Rτ the Reeb vector field of τ , defined by ıRτdτ = 0
and τ(Rτ ) = 1. One among Rτ and −Rτ is a positive section of kerω. We say that
ω is of positive or negative contact type accordingly.
Remark 2.13. If we fix a positive section Z of kerω, being of positive (respec
tively negative) contact type is equivalent to finding τ ∈ Pω such that τ(Z) : N → R
is a positive (respectively negative) function. In this case Rτ = Zτ(Z) .
Example 2.14. If (M, g) is a Riemannian surface as above, then ω′0 = dλ
∣∣
SM
is a HS of positive contact type. Indeed, X is a positive section of kerω′0 and
λ(x,v)(X) = gx(v, d(x,v)piX) = 1. This implies that λ
∣∣
SM
is a positive contact form
and X is its Reeb vector field.
Example 2.15. Suppose that pi : E →M is an S1bundle over an oriented sur
face (M, oM ). If σ ∈ Ω2(M) is a positive symplectic form, then pi∗σ is a Hamiltonian
structure. We claim that pi∗σ is exact if and only if E is nontrivial. To prove neces
sity, we observe that if E is trivial, the integral of pi∗σ over a section of pi is nonzero.
To prove sufficiency, we define e 6= 0 the Euler number of the S1bundle. Then, we
use the classical result [Kob56] saying that a 2form on M , whose integral is 2pie,
is a curvature form for an S1connection on E. Thus, it is exact. This implies that
there exists τ ∈ Ω1(E) such that
• τ(V ) = 1 , • dτ = −2pie
[σ]
pi∗σ . (2.25)
In this case − [σ]2pieτ is a primitive of pi∗σ and a contact form with Reeb vector field
−2pie[σ] V . Therefore, pi∗σ is a Hamiltonian structure of contact type. It is positive if
and only if e is positive.
Example 2.16. For examples of structures of contact type in the context of the
circular planar restricted threebody problem, we refer to [AFvKP12].
2.4. HAMILTONIAN STRUCTURES 25
The most direct way to detect the contact property is to use Remark 2.13.
However, this method could be difficult to apply, especially if we want to prove that
a HS is not of contact type, since we should check that every function τ(Z) vanishes
at some point. This problem is overcome by the following necessary and sufficient
criterion contained in McDuff [McD87].
Proposition 2.17. Let ω′ be an exact HS and Z be a positive section of kerω′.
Then, ω′ is of positive (respectively negative) contact type if and only if the action
of every nullhomologous Zinvariant measure is positive (respectively negative).
We end this subsection by recalling the basic notions about invariant measures
needed in the statement of the proposition above.
If Z is a vector field on a closed manifold N , a Zinvariant measure ζ is a Borel
probability measure on N , such that
∀h : N → R,
∫
N
dh(Z)ζ = 0. (2.26)
We denote the set of Zinvariant measure by M(Z). We associate to every ζ ∈M(Z)
an element ρ(ζ) in H1(N,R)∗ = H1(N,R) defined as
∀ [β] ∈ H1(N,R), < ρ(ζ), [β] > :=
∫
N
β(Z)ζ. (2.27)
Suppose Z is a positive section of kerω, with ω an exact HS on an oriented three
manifold N , ζ ∈M(Z) and τ ∈ Pω′ . The action of ζ is defined as
AτZ : M(Z) −→ R
ζ 7−→
∫
N
τ(Z)ζ. (2.28)
If ζ is nullhomologous (namely ρ(ζ) = 0), then AτZ(ζ) is independent of τ and,
therefore, in this case we write AωZ(ζ) := AτZ(ζ).
2.4.2. The ConleyZehnder index. We finish this section by associating an
integer, called the ConleyZehnder index, to a periodic orbit of a Hamiltonian struc
ture. It encodes information about the linearisation of the system along the orbit
and will play a crucial role in defining the notion of dynamical convexity for HS of
contact type. We refer to [HK99] for proofs and further details.
We start with the definition of the Maslov index for a path with values in Sp(1),
the group of 2× 2symplectic matrices. For any T > 0, we set
SpT (1) := {Ψ : [0, T ]→ Sp(1)  Ψ(0) = Id}.
We call Ψ ∈ SpT (1) nondegenerate if Ψ(T ) does not have 1 as eigenvalue.
Given Ψ ∈ SpT (1), we associate to every u ∈ R2\{0} a winding number ∆θ(Ψ, u)
as follows. Identify R2 with C and let
Ψ(t)u
Ψ(t)u = e
iθΨu (t), (2.29)
26 2. PRELIMINARIES
for some function θΨu : [0, T ]→ R. We define ∆θ(Ψ, u) := θΨu (T )− θΨu (0). Let
I(Ψ) :=
{
1
2pi
∆θ(Ψ, u)
∣∣∣ u ∈ R2 \ {0}} . (2.30)
The interval I(Ψ) is closed and its length is strictly less than 1/2. We notice that
the set e2piiI(Ψ) ⊂ S1 is completely determined by the endpoint Ψ(T ). In particular,
we see that Ψ is nondegenerate if and only if Z∩ ∂I(Ψ) = ∅. We define the Maslov
index for a nondegenerate path as
µ(Ψ) :=
{
2k, if k ∈ I(Ψ), for some k ∈ Z;
2k + 1, if I(Ψ) ⊂ (k, k + 1), for some k ∈ Z. (2.31)
We extend the definition to the degenerate case either by taking the maximal lower
semicontinuous extension µl or the minimal upper semicontinuous extension µu.
This amounts to using the same recipe as in the nondegenerate case, but in the
definition of µl, respectively of µu, we shift the interval I(Ψ) to the left, respectively
to the right, by an arbitrarily small amount. For any k ∈ Z, there hold
µl(Ψ) ≥ 2k + 1 =⇒ I(Ψ) ⊂ (k,+∞) =⇒ µ∗(Ψ) ≥ 2k + 1, (2.32)
µu(Ψ) ≤ 2k − 1 =⇒ I(Ψ) ⊂ (−∞, k) =⇒ µ∗(Ψ) ≤ 2k − 1, (2.33)
where we have used µ∗ to denote both µl and µu.
We move now to describe the ConleyZehnder index for a Hamiltonian structure
ω on (N,O). Consider the 2dimensional distribution ξω :=
TN
kerω and observe that
(ξω, ω) → N is a symplectic vector bundle, where ω is the symplectic form on the
fibres induced by ω. Let ν be a freehomotopy class of loops in N and choose γν a
reference loop in ν together with a symplectic trivialisation Υν of ξω along γν . If
ν is the trivial class, just choose a constant loop with the constant trivialisation.
Suppose that c1(ξω) is νatoroidal. This means that the integral of c1(ξω) over a
cylinder i : C := [0, 1]× T1 → N such that i(0, ·) = γν(·) depends only on i(1, ·).
Let Z be a positive section of kerω and observe that dΦZt acts on (ξω, ω) as a
symplectic bundle map. Let (γ, T ) be a periodic orbit of Z in the class ν. Choose
a cylinder i : C → N connecting γν with γ and let Υ : (i∗ξω, i∗ω) → (2C , ωst) be a
ωsymplectic trivialisation of i∗ξω on C extending Υν . Here 2C is the trivial rank
2vector bundle over C. We form the path of symplectic matrices ΨC,Υγ ∈ SpT (1)
ΨC,Υγ (t) := Υγ(t) ◦ dγ(0)ΦZt ◦Υ−1γ(0) ∈ Sp(1). (2.34)
Definition 2.18. The ConleyZehnder index of γ is µ∗CZ(γ,Υν) := µ
∗(ΨC,Υγ ).
This number does not depend on the choice of Z and the hypothesis on the Chern
class ensures that it is also independent of the pair (C,Υ). If we choose a different
γν with a different trivialisation Υν the index gets shifted by an integer which is the
same for every closed orbit γ in ν.
Remark 2.19. If ω is of contact type, the vector bundle (ker τ, ω
∣∣
ker τ
) is sym
plectic and isomorphic to (ξω, ω). Hence, we can use the former bundle for the
computation of the index.
CHAPTER 3
Symplectic Cohomology and deformations
The material we present in this chapter build up from the work contained in
[Rit10]. On the one hand, it grew out of several fruitful discussions with Alexander
Ritter and it is likely to appear in a forthcoming paper [BR]. On the other hand,
the statement and proof of the crucial Proposition 3.35, which enables us to prove
the main result of this section (Theorem 3.30), are due exclusively to the author of
the present thesis.
The focus is Symplectic Cohomology for general convex manifolds as first deve
loped by Viterbo in [Vit99]. In the next chapter, we will apply the abstract results
proved here to magnetic systems.
In Section 3.1 we recall the notion of convex symplectic manifold (W,ω, j) in
the open and compact case and we give a notion of isomorphism in such categories.
In Section 3.2, we introduce the Symplectic Cohomology of such manifolds, a set of
algebraic invariants SH∗ν counting 1periodic orbits of Hamiltonian flows in the free
homotopy class ν. We will work under the hypothesis that c1(ω) is νatoroidal.
In Section 3.3 we see how to define SH∗ν perturbing autonomous Hamiltonian
functions close to the nonconstant periodic orbits.
In Section 3.4, we define SH∗ν using a subclass of autonomous Hamiltonians,
whose nonconstant periodic orbits are reparametrised Reeb orbits for the contact
form at infinity. For such Hamiltonians we have filtrations on the complex yielding
SH∗ν given by the action (when (W,ω) a Liouville domain) and, in general, by the
period of Reeb orbits. These filtrations will be used for the applications in the next
chapter.
In Section 3.5 we quote the results contained in [Rit10], asserting that SH∗ν
does not depend on the isomorphism class of a convex manifold.
In Section 3.6, we prove the invariance of SH∗ν for compact manifolds under
convex deformations (Ws, ωs, js) which are projectively constant on νtori.
3.1. Convex symplectic manifolds
Let us start by treating the noncompact case. First, we define the manifolds
we are interested in.
Definition 3.1. Let W be an open manifold (noncompact and without bound
ary) of dimension 2m and let Σ be a closed manifold of dimension 2m− 1. We say
that W is an open manifold with cylindrical end (modelled on Σ) if there exists a
diffeomorphism j = (r, p) : U → (a,+∞)×Σ. Here U is an open subset of W , such
that W \ j−1((b,+∞)× Σ) is a compact subset of W with boundary j−1({b} × Σ),
27
28 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
for every (or, equivalently, some) b > a. We write i : Σ → W for the embedding
i := j−1
∣∣
{b}×Σ.
We call j (or less precisely U) a cylindrical end of W and we denote by ∂r ∈ TU
the vector field generated by the coordinate r.
We now fix some additional notation for open manifolds with cylindrical end.
Definition 3.2. If f : Σ→ (a,+∞), define by Γf : Σ→ (a,+∞)× Σ the map
Γf (x) = (f(x), x) and by Σf the image of Γf inside W . Denote by W
f and Wf the
compact and noncompact submanifolds of W with boundary Σf . If we have two
functions f0, f1 : Σ → (a,+∞), with f0 < f1, let W f1f0 be the compact submanifold
between Σf0 and Σf1 .
We proceed to consider the subclass of open manifolds for which the symplectic
structure at infinity is compatible with some contact structure on the model Σ.
Definition 3.3. Let (W,ω) be a symplectic manifold having a cylindrical end
j : U → (a,+∞) × Σ. Suppose that (W,ω) is exact at infinity, meaning that ω
is exact on U . We say that the triple (W,ω, j) is a convex symplectic manifold if
θ := ı∂rω ∈ Ω1(U) is a primitive for ω on U , or, equivalently, if (j−1)∗θ = erp∗α for
some (unique) contact form α on Σ. We denote by Rα ∈ Γ(Σ) the Reeb vector field
of α and we define the auxiliary function ρ := er. We call ∂r the Liouville vector
field and θ the Liouville form.
We say that a convex symplectic manifold is a Liouville domain, if θ extends to
W in such a way that ω = dθ on the whole manifold.
Remark 3.4. Since ω is a symplectic form, an α satisfying the equality above
is automatically a contact form on Σ inducing the orientation obtained from ωn
∣∣
U
putting the vector ∂r first. Using the language introduced in the previous chapter
we can say that every Σf is a hypersurface of positive contact type in (W,ω).
We now define isomorphisms of convex symplectic manifolds.
Definition 3.5. Let (Wi, ωi, ji), for i = 0, 1, be two convex symplectic mani
folds. An isomorphism between (W0, ω0, j0) and (W1, ω1, j1) is a diffeomorphism
F : W0 →W1 with the following properties:
(1) F is a symplectomorphism, i.e. F ∗ω1 = ω0,
(2) there exist open cylindrical ends Vi ⊂ Ui such that F (V0) ⊂ V1,
(3) j1 ◦ F ◦ j−10
∣∣
V0
(r0, p0) = (r0 − f(p0), ψ(p0)) with ψ∗α1 = efα0.
The third condition is equivalent to (F
∣∣
V0
)∗θ1 = θ0. Furthermore, a simple argument
shows that ψ : Σ0 → Σ1 is indeed a diffeomorphism.
Remark 3.6. Let (W,ω) be a symplectic manifold with cylindrical end and
suppose we are given two convex ends j0 and j1. If θ0 = θ1 (or, equivalently,
∂r0 = ∂r1) on some common cylindrical end, then Id : (W,ω, j0) −→ (W,ω, j1) is an
isomorphism. For this reason, we will also use the notation (M,ω, θ) to designate
any convex symplectic manifold (M,ω, j) such that θ := ı∂rω.
3.2. SYMPLECTIC COHOMOLOGY OF CONVEX SYMPLECTIC MANIFOLDS 29
For compact symplectic manifolds we can give an analogous notion of convexity.
Definition 3.7. Let (W,ω) be a compact symplectic manifold with boundary Σ
and let j = (r, p) : U → (−ε, 0]×Σ be a collar of the boundary. We say that (W,ω, j)
is a (compact) convex symplectic manifold if θ := i∂rω ∈ Ω1(U) is a primitive for ω
on U , or equivalently, if (j−1)∗θ = erp∗α for some (unique) contact form α on Σ.
A standard construction called completion yields an open convex symplectic
manifold (Wˆ , ωˆ, jˆ) starting from a compact convex symplectic manifold (W,ω, j):
(Wˆ , ωˆ, jˆ) := (W,ω, j)
⊔
j
(
(−ε,+∞)× Σ, d(erα), Id
)
, (3.1)
The new object is obtained by gluing along j the original convex symplectic manifold
with a positive cylindrical end of the symplectisation of (Σ, α).
Definition 3.8. We say that two convex compact symplectic manifolds are
isomorphic if their completions are isomorphic according to Definition 3.5.
Remark 3.9. Let (W,ω) be a symplectic manifold with convex cylindrical end
j : U → (a,+∞) × Σ. Let f : Σ → (a,+∞) be any function and consider W f (see
Definition 3.2). Then, (W f , ω
∣∣
W f
, j
∣∣
W f
) is a compact convex manifold. We do not
lose any information by passing from W to W f . Indeed, we observe that we can
get back the whole open manifold by taking the completion as explained above. In
other words,
(Wˆ f , ωˆ
∣∣
W f
, jˆ
∣∣
W f
) ' (W,ω, j). (3.2)
We can rephrase (3.2) by saying that every open convex symplectic manifold (W,ω)
is the completion of a compact convex submanifold (W ′, ω
∣∣
W ′) ⊂ (W,ω). This has a
natural generalisation for a family of open convex symplectic manifolds (W,ωs, js).
Namely, there exists a corresponding family of zerocodimensional compact subman
ifolds Ws ⊂W such that ∂Ws is jsconvex and
(Wˆs, ωˆs
∣∣
Ws
, jˆs
∣∣
Ws
) ' (W,ωs, js). (3.3)
3.2. Symplectic Cohomology of convex symplectic manifolds
Assume for the rest of this section that (W,ω, j) is an open convex symplectic
manifold and ν is a free homotopy class of loops in W .
3.2.1. Preliminary conditions. We now define the Symplectic Cohomology
of (W,ω, j) in the class ν under the assumption that
(S1) all the Reeb orbits of the associated contact form α in the classes i−1∗ (ν)
are nondegenerate (in this case we say that α is νnondegenerate);
(S2) c1(ω) is νatoroidal.
Denote by Spec(α, ν) the set of periods and by T (α, ν) > 0 the minimal period of
a Reeb orbit of α in the class ν. Assumption (S1) guarantees that Spec(α, ν) is a
discrete subset of [T (α, ν),+∞).
30 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
3.2.2. Admissible Hamiltonians. Symplectic Cohomology counts the num
ber of 1periodic orbits in the class ν for a particular kind of 1periodic Hamiltonians.
To introduce them, we need first the following two general definitions.
Definition 3.10. Let (W,ω) be a symplectic manifold and consider a function
H : T1 ×W → R. We say that a 1periodic orbit x for XH is nondegenerate if 1
does not belong to the spectrum of dx(0)Φ
H
1 . We say that H is νnondegenerate, if
all the 1periodic orbits in the class ν are nondegenerate.
Definition 3.11. Let (W,ω, j) be a symplectic manifold with cylindrical end.
A function H : T1 ×W → R is said to have constant slope at infinity, if there exist
constants TH ∈ (0,+∞) and aH ∈ R such that H ◦ j−1 = THer + aH on some
cylindrical end V ⊂ U . The number TH is called the slope of H.
We are now ready to introduce the class of Hamiltonians that we are going to
use on (W,ω, j).
Definition 3.12. A function H : T1×W → R is called νadmissible if it satisfies
the following two properties:
(H1) it is νnondegenerate,
(H2) it has constant slope at infinity and TH does not belong to Spec(α, ν).
We denote the set of all νadmissible Hamiltonians by Hν .
3.2.3. The action 1form, the grading and the moduli spaces. Consider
H ∈ Hν . Its 1periodic orbits are the zeroes of a closed 1form dAωH on LνW . If
x ∈ LνW and ξ ∈ Γ(x∗TW ) is an element in Tx(LνW ), the 1form is defined by
dxAωH · ξ =
∫
T1
(
ıx˙(t)ωx(t) + dx(t)H(t, x(t))
) · ξ(t) dt. (3.4)
The action 1form yields a function AωH on the set of νcylinders by integration.
If u : [a, b]→ LνW is a νcylinder, we set
AωH(u) : =
∫ b
a
du(s)AωH ·
du
ds
(s) ds
= −ω(u) +
∫
T1
H(t, u(b, t)) dt−
∫
T1
H(t, u(a, t)) dt,
where ω(u) is the integral of ω over u. Observe that AωH(u) does not change if we
homotope the cylinder u keeping the end points fixed.
We can associate a degree x to every 1periodic orbit x in ν as follows. Choose
a reference loop xν in the class ν and fix a symplectic trivialisation Υxν of x
∗
ν(TW ).
Take a connecting cylinder Cx from xν to x and extend the trivialisation over Cx.
This will induce a symplectic trivialisation Υx of x
∗TW , whose homotopy class
depends only on Υxν since c1(ω) is νatoroidal. Writing the linearisation of the
Hamiltonian flow dΦXHt using Υx yields a path of symplectic matrices along x.
We call ConleyZehnder index the Maslov index of this path and we denote it by
µCZ(x) ∈ Z. Finally, we set x := dim(W )2 − µCZ(x).
3.2. SYMPLECTIC COHOMOLOGY OF CONVEX SYMPLECTIC MANIFOLDS 31
We now define a moduli space of cylinders connecting two 1periodic orbits using
a particular kind of almost complex structures compatible with convexity.
Definition 3.13. Let (W,ω, j) be a convex manifold. An ωcompatible 1
periodic almost complex structure J is convex if for big r, J is independent of
time and (dρ) ◦ J = −θ.
Take a 1periodic ωcompatible convex almost complex structure J and consider
Floer’s equation for cylinders u : R× T1 →W
∂su+ J(∂tu−XH) = 0 . (3.5)
If u is a Floer trajectory we denote its energy by
E(u) :=
∫
R
‖∂su‖2J ds =
∫
R×T1
∂su2J ds dt (3.6)
Using Floer’s equation (3.5), one gets the identity
E(u) = −AωH(u). (3.7)
Call M′(H,J, x−, x+) the space of Floer trajectories that converge uniformly
to the 1periodic orbits x± for s → ±∞. Suppose that J is Hregular, namely
that the operator u 7→ ∂su + J(∂tu − XH) on the space of all cylinders is regular
at M′(H,J, x−, x+). This implies that all the moduli spaces are smooth manifolds
and, if nonempty, dimM′(H,J, x−, x+) = µCZ(x+)− µCZ(x−) = x− − x+.
Let M(H,J, x−, x+) be the quotient of M′(H,J, x−, x+) under the Raction
obtained by shifting the variable s. If A is a homotopy class of cylinders relative
ends, we denote byMA(H,J, x−, x+) the subset of the moduli space whose elements
belong to A.
3.2.4. The cochain complex and the differential. We build a complex
SC∗ν (W,ω, j,H) as the free Λmodule generated by the 1periodic orbits x of H.
Here Λ is the Novikov ring defined as
Λ :=
{
+∞∑
i=0
nit
ai
∣∣ ni ∈ Z, ai ∈ R, lim
i→+∞
ai = +∞
}
.
The differential δJ : SC
∗
ν (W,ω, j,H)→ SC∗+1ν (W,ω, j,H) of such complex is defined
on the generators as
δJx :=
∑
u∈M(H,J,y,x)
y−x=1
(u) t−A
ω
H(u)y,
where (u) is an orientation sign defined in [Rit13, Appendix 2]. We extend it on
the whole SC∗ν (W,ω, j,H) by Λlinearity. If x and y are two periodic orbits, we
denote by < δJx, y >, the component of δJx along the subspace generated by y.
By a standard argument in Floer theory, in order to prove that δJ is a well
defined map of Λmodules and that δJ ◦ δJ = 0, we need to prove that the elements
of MA(H,J, x−, x+) have:
a) Uniform C0bounds; these stem from a maximum principle.
32 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
b) Uniform C1bounds; these follow from Lemma 3.14 below, where we show that
c1(ω) is aspherical (see [HS95]).
c) Uniform energy bounds; these follow readily from (3.7).
Lemma 3.14. If σ ∈ Ω2(W ) is νatoroidal, then σ is aspherical.
Proof. We have an action ] : pi2(W ) × H2(W,Z) → H2(W,Z) by connected
sum. Moreover, if ([S2], [M ]) ∈ pi2(W )×H2(W,Z), then
[σ]([S2]M ]) = [σ]([S2]) + [σ]([M ]).
Therefore, the lemma is proven if we show that the action preserves classes repre
sented by νtori. Indeed, in this case the above identity would become
0 = [σ]([S2]) + 0.
Suppose M is a νtorus parametrised by (s, t) 7→ ΓM (s, t) = γMs (t), with γs ∈ ν and
(s, t) ∈ T1 × T1. The connected sum between M and some S2 can be represented
by a map ΓS
2]M : T1 × T1 → W , which concides with ΓM outside a small square
[s0, s1]× [t0, t1] and inside the square it coincides with the map
[s0, s1]× [t0, t1] ∼−→ S2 \D2 −→W,
where D2 is the small disc that we remove from S2 in order to perform the connected
sum. Hence, S2]M is still a torus and the curves t 7→ γS2]Ms (t) := ΓS2]M (s, t) are
still in the class ν because γS
2]M
s = γMs if s /∈ [s0, s1].
Thanks to the lemma, we have welldefined cohomology groups obtained from
δJ . We denote them by SH
∗
ν (W,ω, j,H, J). We put a partial order on pairs (H,J),
where H and J are as above, by saying that (H+, J+) (H−, J−) if and only if
TH+ ≤ TH− (we use the reverse notation for the signs, since in the definition of maps
on a generator x, we take the moduli space of cylinders arriving at x).
When (H+, J+) (H−, J−) we can construct continuation maps
ϕH
+,J+
H−,J− : SH
∗
ν (W,ω, j,H
+, J+) −→ SH∗ν (W,ω, j,H−, J−)
such that,
(H+, J+) (H0, J0) (H−, J−) =⇒ ϕH0,J0
H−,J− ◦ ϕH
+,J+
H0,J0
= ϕH
+,J+
H−,J− .
Such maps are defined as follows. We consider a homotopy (Hs, Js) with s ranging
in R, which is equal to (H−, J−), for s very negative, and to (H+, J+), for s very
positive. Such homotopy yields an sdependent Floer equation
∂su+ J
s(∂tu−XHs) = 0 . (3.8)
As before, we denote by MA({Hs}, {Js}, x−, x+) the moduli spaces of solutions
in the class A, connecting a 1periodic orbit of XH− with a 1periodic orbit of
XH+ . They are smooth manifolds and, when they are nonempty, we have that
dimMA({Hs}, {Js}, x−, x+) = µCZ(x+)−µCZ(x−) = x− − x+. For these moduli
spaces we have to show properties a), b), c) as before. Property a) stems again from
3.3. PERTURBING A NONDEGENERATE AUTONOMOUS HAMILTONIAN 33
the maximum principle, which holds provided ddsTHs ≤ 0. Property b) is once more
a consequence of Lemma 3.14. Property c), stems from the modified energyaction
identity
E(u) = −AωH+(u) +
∫
T1
(H− −H+)(t, x−(t))dt+
∫
R×T1
(∂sH
s)(t, u(s, t))dsdt , (3.9)
which replaces (3.7) for sdependent cylinders. Notice that the rightmost term of
(3.9) is bounded thanks to the uniform C0bounds and the fact that ∂sH
s 6= 0 only
on a finite interval. If we define η(x−) :=
∫
T1(H
+ − H−)(t, x−(t))dt, then at the
chain level
ϕH
+,J+
H−,J−x :=
∑
u∈M({Hs},{Js},y,x)
y−x=0
(u) tη(y)−A
ω
H+
(u)y . (3.10)
An argument similar to the one at the end of Section 3.6.5, shows that such a map
intertwines δJ− and δJ+ , and, therefore, yields the desired map in cohomology.
The Symplectic Cohomology (in the class ν) is defined to be the direct limit of
the direct system
(
SH∗ν (W,ω, j,H, J), ϕ
H+,J+
H−,J−
)
:
SH∗ν (W,ω, j) := lim−→
(H,J)
SH∗ν (W,ω, j,H, J).
We define the Symplectic Cohomology of a compact convex symplectic manifold
as the Symplectic Cohomology of its completion: SH∗ν (W,ω, j) := SH∗ν (Wˆ , ωˆ, jˆ).
3.3. Perturbing a nondegenerate autonomous Hamiltonian
We saw that the Symplectic Cohomology of (W,ω, j) is defined starting from
a Hamiltonian function whose 1periodic orbits are nondegenerate. However, if
H : W → R is an autonomous Hamiltonian and x is a nonconstant 1periodic
orbit for the flow of XH , x belongs to an S
1worth of periodic orbits. Hence, x is
degenerate. In this setting, we have to look to a weaker notion of nondegeneracy.
Definition 3.15. Let x be a 1periodic orbit of XH , where H is an autonomous
Hamiltonian and denote by Sx the connected component of set of 1periodic orbits
of XH to which x belongs. We say that x is transversally nondegenerate
• either if x is a constant orbit and it is nondegenerate according to Definition
3.10 (namely, dx(0)Φ
XH
1 does not have 1 in the spectrum);
• or if x is nonconstant and 1 has algebraic multiplicity 2 in the spectrum
of dx(0)Φ
XH
1 .
We say that H is transversally νnondegenerate if all 1periodic orbits in the class
ν are transversally nondegenerate.
Remark 3.16. When dimW = 4 the notion of transversally nondegenerate
coincides with the notion given for Hamiltonian structures in Definition 2.10.
Remark 3.17. If x is transversally nondegenerate, then Sx = {x(· + t′)}t′∈T1 .
Therefore, Sx = {x}, if x is constant and Sx ' T1, if x is not constant.
34 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
If H is an autonomous Hamiltonian such that all the 1periodic orbits of XH in
the class ν are transversally nondegenerate, we can construct a small perturbation
Hε : W × S1 → R such that
• Hε differs from H only in a small neighbourhood Uε of the nonconstant
periodic orbits of XH and ‖H −Hε‖ is small;
• the constant periodic orbits of XHε are the same as the constant periodic
orbits of XH ;
• for every nonconstant 1periodic orbit x of XH there are two periodic orbits
xmin and xMax of XHε supported in Uε. All the nonconstant periodic orbits
of XHε arise in this way.
Consider a function H : W → R that satisfies the following two properties:
(H’1) it is transversally νnondegenerate,
(H’2) it has constant slope at infinity and TH does not belong to Spec(α, ν) .
Denote the set of such functions by H′ν . We readily see that if we carry out the above
perturbation to H ∈ H′ν , the resulting function Hε belongs to Hν (see Definition
3.12) and we can use it to compute Symplectic Cohomology.
The degree of the new nonconstant orbits is given by
• xmin = dim(W )
2
− µCZ(x), • xMax = dim(W )
2
− µCZ(x)− 1, (3.11)
where µCZ(x) is the transverse ConleyZehnder index. It will be interesting for the
applications to compute < δJxMax, xmin >. Proposition 3.9(ii) in [BO09b] tells us
that
< δJxMax, xmin > =
0 if x is good,±2tax if x is bad, (3.12)
where ax is a small positive number. Recall that an orbit is bad if it is an even
iteration of a hyperbolic orbit with odd index and it is good otherwise.
3.4. Reeb orbits and two filtrations of the Floer Complex
In this section we restrict the admissible Hamiltonians to a subclass Hˆν ⊂ Hν ,
whose nonconstant periodic orbits are in strict relation with Reeb orbits of α.
Definition 3.18. Fix some b belonging to the image of the function r : U → R
and denote ρb := e
b. Fix also a C2small Morse function Hb : W
b → R, such that
close to the boundary Hb = hb(e
r) for some strictly increasing convex function hb.
Let H : W → R be an element of H′ν with the following additional properties
(Hˆ1) on W b, H = Hb,
(Hˆ2) on Wb, H = h(e
r) for a function h : [ρb,+∞)→ which is strictly increasing
and strictly convex on some interval [ρb, ρH) and satisfies h(ρ) = THρ+aH ,
for some TH /∈ Spec(α, ν) and aH ∈ R, on [ρH ,+∞).
We denote by Hˆ′ν the subset of all the Hamiltonians in H′ν satisfying these two
properties and by Hˆν ⊂ Hν the corresponding set of perturbations inside Hν .
3.4. REEB ORBITS AND TWO FILTRATIONS OF THE FLOER COMPLEX 35
The 1periodic orbits of H ∈ Hˆ′ν fall into two classes: constant periodic orbits
inside W b and nonconstant periodic orbits inside Wb. Observe that in the region
Wb, XH = h
′Rα where h′ is the derivative of h. Hence, x(t) = (r(t), p(t)) is a
1periodic orbit for XH if and only if r(t) is constant and equal to some log ρx and
there exists a h′(ρx)periodic Reeb orbit γ : TT → Σ such that p(t) = γ(h′(ρx)t).
Since h′ is increasing and h′(ρb) < T (α, ν), the nonconstant periodic orbits of XH
are in bijection with all the Reeb orbits with period smaller than TH .
Consider now the Floer Complex (SC∗ν (W,ω, j,Hε), δJ) with Hε ∈ Hˆν and some
J which is Hεadmissible. In the next subsections we show that we have two ways
of filtrating this complex.
The first one is a filtration by the action and it works when (W,ω) is a Liouville
domain. In this case we can define the action functional AωHε : LνW → R on the
space of loops in the class ν.
The second one is a filtration by the period of Reeb orbits. It works for general
convex manifolds but we have to restrict the class of admissible complex structures
to some J (H) (see Definition 3.22), so that we can apply the maximum principle
(see [BO09a, page 654] and Lemma 3.23).
When ν = 0, both filtrations imply that the singular cohomology complex
(C∗+n(W,Λ), δ0) shifted by n is a subcomplex of (SC∗ν (W,ω, j,Hε), δJ).
3.4.1. Filtration by the action for Liouville domains. Suppose (W,ω) is
a Liouville domain. This means that the Lioville form θ extends to a global primitive
for ω on the whole W . If H : T1 ×W → R is any Hamiltonian function, we define
the action functional AωH : LνW → R by
AωH(x) = −
∫
T1
x∗θ +
∫
T1
H(t, x(t))dt ,
If u is a Floer cylinder connecting x− and x+, (3.7) yields
0 ≤ E(u) = −AωH(u) = AωH(x−)−AωH(x+) .
This implies that AωH(x−) ≥ AωH(x+). Therefore, if H is νnondegenerate and J
is Hadmissible, then δJ preserves the superlevels of AωH and we get the following
corollary.
Corollary 3.19. If x and y are two different 1periodic orbits for H and
AωH(y) ≤ AωH(x), then
< δJx, y >= 0 . (3.13)
We denote by SC∗,>aν (W,ω, j,H) the subspace of SC∗ν (W,ω, j,H) generated by
all the orbits with action bigger than some a ∈ R. Corollary 3.19 shows that δJ
restricts to a linear operator on this subspace, and the inclusion induces a map in
cohomology:
SH∗,>aν (W,ω, j,H, J) −→ SH∗ν (W,ω, j,H, J) . (3.14)
Let (H+, J+) (H−, J−) and let (Hs, Js) be a homotopy between the two pairs
satisfying ddsTHs ≤ 0. If u is an sdependent Floer cylinder, relation (3.9) can be
36 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
rewritten as
AωH+(x+)−AωH−(x−) = −E(u) +
∫
R×T1
(∂sH
s)(t, u(s, t))dsdt .
From this equation we see that the continuation map preserves the filtration, pro
vided that ∂sH
s ≤ 0. Under this additional hypothesis, ϕH+,J+
H−,J− induces a map
SH∗,>aν (W,ω, j,H
+, J+) −→ SH∗,>aν (W,ω, j,H−, J−) . (3.15)
Let us now specialise to Hamiltonians H ∈ Hˆ′ν . Call ah : [ρb,+∞) → R, the
function ah(ρ) := h(ρ)− ρh′(ρ). The absolute value of ah(ρb) is small and
a′h(ρ) = −ρh′′(ρ) . (3.16)
Hence, ah is strictly decreasing on [ρb, ρH ] and ah ≡ aH on [ρH ,+∞).
If x is a nonconstant periodic orbit in the class ν, then x ⊂ Wb and we can
write x(t) = (log ρx, γx(h
′(ρx)t)), for some Reeb orbit γx. We compute
AωH(x) = −
∫ h′(ρx)
0
ρxγ
∗
xα + h(ρx) = ah(ρx) . (3.17)
Fix a ∈ R and take H+ and H− in Hˆ′0, such that TH+ ≤ TH− and H− = H+ on
some W c, where a = ah+(e
c) = ah−(e
c). In particular, a > max{aH+ , aH−}. Take
perturbations H+ε and H
−
ε and let (H
s
ε , J
s) be a homotopy between them such that
∂sH
s
ε ≤ 0 and which is constant on W c. We readily see that
SC∗,>aν (W,ω, j,H
+
ε ) = SC
∗,>a
ν (W,ω, j,H
−
ε ) .
Consider continuation cylinders between the generators of these two subcomplexes.
The maximum principle shows that they are contained inside W c. However, in this
region ∂sH
s = 0 and the transversality of the moduli spaces shows that we only
have the constant cylinders. We have proved the following corollary.
Corollary 3.20. Under the hypotheses of the paragraph above, the filtered con
tinuation map SC∗,>aν (W,ω, j,H+ε )→ SC∗,>aν (W,ω, j,H−ε ) is the identity.
Take now ν = 0 and assume that H ∈ Hˆ′0. We want to use the map (3.14) to
single out the constant orbits of H among all contractible orbits. If x is a constant
orbit, we readily compute
AωH(x) = H(x) ≥ min
W b
H . (3.18)
If x(t) = (log ρx, γx(h
′(ρx)t)) is nonconstant, we know that AωH(x) = ah(ρx). More
over, (3.16) implies that a′h ≤ −ρbh′′ and, therefore,
ah(ρx) ≤ ah(ρb)− ρb(h′(ρx)− h′(ρb)) .
Since h′(ρx) ≥ T (α, ν) > 0, we see that there exists a0 < 0 such that
AωH(x) < a0 < AωH(x) .
Taking a small perturbation Hε ∈ Hˆ0, this inequality still holds and we find that
SC∗,>a00 (W,ω, j,Hε) is generated by the constant periodic orbits only. A further
3.4. REEB ORBITS AND TWO FILTRATIONS OF THE FLOER COMPLEX 37
analysis analogous to the case of closed symplectic manifolds shows that δJ re
stricted to this subcomplex is the differential for the Morse Cohomology of the
function HεW b = Hb (see [Vit99, Section 1.2]), where we identify constant periodic
orbits of XHε with critical points of Hε up to a shifting of degrees by n. Thus,
we have SH∗,>a00 (W,ω, j,Hε, J) ' H∗+n(W,Λ). If (H+ε , J+) (H−ε , J−) and H+,
H− belong to Hˆ′0, then SH∗,>a00 (W,ω, j,H+ε , J+) ' SH∗,>a00 (W,ω, j,H−ε , J−), by
Corollary 3.20. Since Hˆ0 is cofinal in H0, we can take direct limits and arrive to the
following corollary.
Corollary 3.21. If (W,ω, j) is a Liouville domain, then, for every ∗ ∈ Z, there
exists a map
H∗+n(W,Λ) −→ SH∗0 (W,ω, j) . (3.19)
If α does not have any Reeb periodic orbit γ contractible in W such that
µCZ(γ) ∈ { dimW/2− ∗ , dimW/2− ∗+ 1 } ,
then such map is an isomorphism.
3.4.2. Filtration by the period. Let (W,ω) be a general open convex mani
fold. We restrict the class of admissible almost complex structures as follows.
Definition 3.22. Fix an ωcompatible almost complex structure J0 such that
(dρ) ◦ J0 = −θ on Wb. For every H ∈ Hˆ′ν let J (H) be the set of the ωcompatible
almost complex structures Jε which are small compact perturbations of J0 and which
are Hadmissible for some Hε ∈ Hˆν .
The next lemma, which is taken from [BO09a, page 654], gives us some infor
mation on the behaviour of Floer cylinders for (H,J0) which are asymptotic to a
reparametrisation of a Reeb orbit for s→ −∞.
Lemma 3.23. Take H ∈ Hˆ′ν and suppose that u : (−∞, s∗]× T1 →Wb is a non
constant Floer halfcylinder for the pair (H,J0). If u is asymptotic for s → −∞
to x(t) = (rx, γx(h
′(erx)t)), a nonconstant 1periodic orbit for XH , then u is not
contained in W rx.
Proof. Using the cylindrical end j = (r, p), we write u(s, t) = (r(s, t), p(s, t)).
If we employ the splitting TWb '< ∂r > ⊕ < Rα > ⊕ kerα, the Floer equation for
u takes the following form:
∂sr − α(∂tp) + h′(er) = 0
α(∂sp) + ∂tr = 0
pikerα(∂sp) + J0pikerα(∂tp) = 0 ,
(3.20)
where pikerα is the projection on the third factor of the splitting.
Now define the real functions r and a on the interval (−∞, s∗] by integrating in
the tdirection:
r(s) :=
∫
T1
r(s, t)dt , a(s) :=
∫
T1
p(s, ·)∗α .
38 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
By Stokes’s Theorem we have that
a(s1)− a(s0) =
∫
[s0,s1]×T1
p∗dα . (3.21)
Since J0 leaves kerα invariant and it is ωcompatible, the third equation yields
that the rightmost term in (3.21) is nonnegative and, therefore, a is a nondecreasing
function.
We integrate the first equation of (3.20) in t and use this monotonicity property:
d
ds
r(s) ≥ −
∫
T1
h′(er(s,t))dt+ a(−∞) = h′(erx)−
∫
T1
h′(er(s,t))dt . (3.22)
Suppose now, by contradiction that r(s, t) ≤ rx for every (s, t) ∈ (−∞, s∗] × T1.
Since ρ 7→ h′(ρ) is increasing, we find that the rightmost term in (3.22) is non
negative. Hence, ddsr(s) ≥ 0. We have two possibilities. Either there exists a point
s∗∗ < s∗ such that ddsr(s∗∗) > 0, or
d
dsr ≡ 0 on the whole cylinder. In the first
case, we must have r(s∗∗) > rx, which is a contradiction. In the second case, r ≡ rx
and, since r(s, t) ≤ rx, we see that r(s, t) ≡ rx on the whole cylinder. By the first
equation in (3.20) a is also constant and by (3.21), we have that∫
(−∞,s∗]×T1
p∗dα = 0 .
Putting this together with the fact that the function r(s, t) is constant, we get that
E(u) = 0, which contradicts the fact that u is not constant. Since both cases led to
a contradiction, we see that u is not contained in W rx and the lemma is proven.
We have the following immediate corollary.
Corollary 3.24. Suppose u : R × T1 → W is a nonconstant Floer cylinder
for the pair (H,J0), with H ∈ Hˆ′ν , which connects two 1periodic orbits for XH . If
x− = (rx− , γ−(T−t)) for some T−periodic Reeb orbit γ−, with T− = h′(e
rx− ), then
x+ = (rx+ , γ+(T+t)) for some T+periodic Reeb orbit γ+, with T+ = h
′(erx+ ), such
that
T+ > T− . (3.23)
Proof. Consider the function r(s, t) := r(u(s, t)). By the lemma we just proved,
max r > rx− . By the maximum principle, x+ is not contained in W
rx− . Hence,
x+ = (rx+ , γ+(T+t)) is a nonconstant periodic orbit and rx+ > rx− . The thesis
follows since ρ 7→ h′(ρ) is strictly increasing in the interval [ρb, ρH ].
The previous corollary gives important information on SHν , once we take a
perturbation in order to achieve transversality.
Corollary 3.25. Given H ∈ Hˆ′ν , there exist Hε ∈ Hˆν and Jε ∈ J (H) such
that the following statement is true. Let x and y be 1periodic orbits of XH with
Sx 6= Sy such that x is associated with a Reeb orbit of period Tx and y is either
constant or it is associated with a Reeb orbit of period Ty, with Ty ≤ Tx. If xε and
yε are 1periodic orbits of XHε close to x and y, then
< δJεxε, yε > = 0 . (3.24)
3.4. REEB ORBITS AND TWO FILTRATIONS OF THE FLOER COMPLEX 39
Proof. We argue by contradiction and suppose that there exists εk → 0 and a
sequence of cylinders uk ∈M′(Hεk , Jεk , xεk , yεk).
We claim that, up to taking a subsequence, there is an m ∈ N such that for
every i = 1, . . . ,m there exists a shift sik ∈ R and for every i = 0, . . . ,m there exists
a 1periodic orbit xi for XH with the following properties:
• x0 = x and xm = y,
• for every i = 1, . . . ,m− 1, sik < si+1k ,
• for every i = 1, . . . ,m, uk(· + sik) → ui∞, in the C∞loctopology, for some
cylinders ui∞ satisfying the Floer Equation (3.5) for the pair (H,J0) and
with asymptotic conditions
ui∞(−∞) ∈ Sxi−1 , ui∞(+∞) ∈ Sxi . (3.25)
The claim follows from Proposition 4.7 in [BO09b] with the only difference that in
our case also the complex structure is allowed to vary with k. This however does not
represent a problem, since such proposition relies on Floer’s compactness theorem
[Flo89, Proposition 3(c)], in which the almost complex structure is allowed to vary
as well.
Using inductively Corollary 3.24 and the fact that the positive end of ui−1∞ co
incides with the negative end of ui∞, we see that xi is a nonconstant orbit with
xi(t) = (ri, γi(Tit)) and that Ti ≤ Ti+1. Since Sx 6= Sy, there exists a nonconstant
ui∗∞. This implies that Ti∗−1 < Ti∗ and, therefore, that Ty > Tx. This contradiction
proves the corollary.
As happened for the action filtration, Corollary 3.25 shows that for any T > 0,
δJ leaves invariant the subspace SC
∗
ν, 0 and take H+ and H− in Hˆ′0, such that TH+ ≤ TH− and H− = H+
on some W c, where T = (h+)′(ec) = (h−)′(ec). In particular, T < min{TH+ , TH−}.
Take perturbations H+ε and H
−
ε and let (H
s
ε , J
s
ε ) be a homotopy such that ∂sH
s
ε ≤ 0
and which is constant on W c. We readily see that
SC∗ν, 0, the set of smooth loops
L = {x ⊂ U cH  dxAωsH Js ≤ C, for some s and Js ∈ J {ωs}H }
is relatively compact for the C0topology. By the Arzela´Ascoli Theorem, the claim
follows by showing that the elements of L are equicontinuous with respect to the
distance induced by some Js∗ . We compute
C ≥ dxAωsH Js =
(∫
T1
x˙−XH(x)2Jsdt
) 1
2
≥
(∫
T1
x˙2Jsdt
) 1
2
− ‖XH‖Js .
By (3.33), ‖XH‖Js ≤ C ′ uniformly in Js. Thus, the L2norm of x˙ with respect to
Js is uniformly bounded and, hence, x is uniformly 1/2Ho¨lder continuous for the
distance induced by Js. Since all these distances are Lipschitz equivalent thanks to
(3.33) again, the equicontinuity follows.
Now, let us prove the lemma. Arguing by contradiction, we can find a sequence
of loops (xn), such that
lim
n→+∞ dxnA
ωsn
H Jsn = 0, xn 6⊂ U cH ∩ supp{σs}c = VH . (3.35)
By the claim, passing to a subsequence, which we denote in the same way, xn
converges in the C0norm to x∞ 6⊂ VH . By compactness of the interval, we can
3.6. INVARIANCE UNDER DEFORMATIONS PROJECTIVELY CONSTANT ON TORI 47
also suppose that sn → s∞. We claim that x∞ is differentiable and it is a critical
point of dAωs∞H . Indeed, XωsnH (xn) → Xωs∞H (x∞) in the L2topology. By (3.35),
x˙n → Xωs∞H (x∞) in the L2topology. Thus, x∞ is in W 1,2 and its weak derivative is
X
ωs∞
H (x∞). Since X
ωs∞
H (x∞) is continuous, we conclude that x˙∞(t) exists at every
t and x˙∞(t) = X
ωs∞
H (x∞(t)). Therefore, x∞ is a periodic orbit of H, contradicting
the fact that x∞ 6⊂ VH .
3.6.3. Subdividing the deformation in substeps. We decompose the de
formation ωs in shorter perturbations by a suitable smooth change of parameters
ss0s1 : R→ [s0, s1], such that
ss0s1(S) =
{
s0 if S ≤ 0
s1 if S ≥ 1.
We set ωs0,s1S := ωss0s1 (S)
. Notice that there exists C > 0 such that, if Js ∈ J {ωs},
then ∥∥ω˙s0,s1S ∥∥Js ≤ Cs1 − s0. (3.36)
To ease the notation, we drop the superscript and we write ωS for ω
s0,s1
S . More
over, we denote s0 by − and s1 by +.
3.6.4. Two kind of homotopies and two kind of homotopies of homo
topies. Let H ∈ H{ωs} and let (H,J−) ∈ P− and (H,J+) ∈ P+. A symplectic
homotopy or shomotopy S 7→ (ωS , H, JS) is a path connecting (H,J−) to (H,J+)
and constant at infinity.
Given S ∈ R and (H+, J+S ) (H−, J−S ) in PS , a monotone homotopy or m
homotopy is a path r 7→ (Hr, JrS), with r ∈ R, connecting (H−, J−S ) to (H+, J+S )
and constant at infinity, such that
• each Hr has constant slope at infinity;
• ∂rTHr ≤ 0;
• (JrS) is a family of ωScompatible convex almost complex structures satis
fying bounds analogous to the ones described in condition ii) and iii) of
Section 3.6.1.
Let H− and H+ be in H{ωs} and let J−− , J−+ , J+− , J++ be almost complex struc
tures such that (H+, J+− ) (H−, J−− ) in P− and (H+, J++ ) (H−, J−+ ) in P+. A
symplecticmonotone homotopy or smhomotopy is a map (S, r) 7→ (ωS , Hr, JrS) with
(S, r) ∈ R2 such that
• restricting it to r = ±∞, we get two shomotopies;
• restricting it to S = ±∞, we get two mhomotopies.
Consider a path r 7→ σrS of 2forms constant at infinity and such that σ−S = 0 and
σ+S = σ
+. Suppose that all the elements of the 2parameters family ωrS := ω + σ
r
S
satisfy the hypotheses of Proposition 3.35. We can define a class H{ωrS} of admissible
Hamiltonians associated to {ωrS} in the same way we definedH{ωs} for a 1parameter
deformation. As before, this class yields a set of admissible pairs PrS , which enjoys
properties similar to i), ii), iii) outlined before for Ps. Notice that when S is
48 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
sufficiently big, PrS = P± does not depend on r (and on S). Given (H,J−) ∈ P− and
(H,J+) ∈ P+, a symplecticsymplectic homotopy or sshomotopy is the 2parameter
family (ωrS , H, J
r
S) such that J
r
S = J± for big S and such that restricting it to
r = ±∞, we get two shomotopies.
We remark that the PalaisSmale lemma still holds for sm and sshomotopies.
3.6.5. Moduli spaces for shomotopies and chain maps. We know that
we can associate chain maps called continuation maps to monotone homotopies by
considering suitable moduli spaces. We now carry out a similar construction to
associate chain maps to symplectic homotopies (ωS , H, JS) as well. For this we will
need first to have s1 − s0 < δH , for some δH > 0. The general definition of the
chain map will then be as the composition of intermediate maps as we did before in
(3.30). More precisely, we choose δH in such a way that
µH := sup
(S,Js)∈R×J {ωs}
∥∥ω˙s1,s0S ∥∥Js ≤ CδH ≤ 1εK
εH
‖XK‖+ 1 , (3.37)
where we have defined
‖XK‖ := sup
Js∈J {ωs}
sup
z∈supp{σs}
XK(z)Js .
Observe that ‖XK‖ is finite thanks to condition ii) in Section 3.6.1.
Thus, let (ωS , H, JS) be a symplectic homotopy satisfying (3.37) and define
the moduli spaces M(H, {JS}, x−, x+) of Sdependent Floer cylinders relative to
such homotopy, connecting two 1periodic orbits x− and x+ of H. Choosing the
family {JS} in a generic way, we can suppose that M(H, {JS}, x−, x+) is a smooth
manifold. If this space is nonempty, we have
dimM(H, {JS}, x−, x+) = µCZ(x+)− µCZ(x−) = x− − x+ .
If A is a homotopy class of cylinders relative ends, let MA(H, {JS}, x−, x+) be the
subset of the moduli space whose elements belong to A.
We claim that MA(H, {JS}, x−, x+) is relatively compact in the C∞loctopology.
As before, this is achieved in a standard way once we have the three ingredients
below.
a) Uniform C0bounds. They stem from the fact that the perturbation σS is com
pactly supported and, hence, the maximum principle still applies.
b) Uniform C1bounds. As before, they stem from the fact that bubbling off of
holomorphic spheres generically does not occur by Lemma 3.14.
c) Uniform bounds on the energy E as defined in (3.6). These require a longer
argument, which we describe below.
First, observe that Lemma 3.37 allows to bound the amount of time an S
dependent Floer cylinder u in MA(H, {JS}, x−, x+) spends on supp{σs}. Define
S(u,H) := {S ∈ R u(S) ∩ supp{σs} 6= ∅},
where we have denoted by u(S) the loop t 7→ u(S, t).
3.6. INVARIANCE UNDER DEFORMATIONS PROJECTIVELY CONSTANT ON TORI 49
Lemma 3.38. The following estimate holds
S(u,H) ≤ E(u)
ε2H
,
where  ·  denotes the Lebesgue measure on R.
Proof. Observe that S(u,H) × T1 ⊂ {(S, t)  ∂SuJS ≥ εH} and then use
Chebyshev’s inequality.
We claim that for an Sdependent Floer cylinder u, identity (3.7) must be re
placed with the more general
E(u) = −Aω+H (u) +
∫
R
ω˙S(uZS ) dS, (3.38)
where ZS := (−∞, S]× T1. Indeed,
E(u) = −
∫
R
du(S)AωSH ·
du
dS
(S) dS = −
∫
R
[
d
dS
(AωSH (uZS ))− (∂AωSH∂S
)
(uZS )
]
dS
= −Aω+H (u) +
∫
R
(
∂AωSH
∂S
)
(uZS ) dS .
The claim follows by observing that
(
∂AωSH
∂S
)
(uZS ) = ω˙S(uZS ).
The first summand on the right of (3.38) depends only on the homotopy class
of u relative ends, so we need to bound only the second one:∫
R
ω˙S(uZS ) dS =
∫ 1
0
ω˙S(uZS )dS ≤ sup
S∈[0,1]
ω˙S (uZS ) ,
where we have used the fact that ω˙S = 0 for S /∈ [0, 1]. We now estimate the last
term, remembering that, for every S ∈ [0, 1], ω˙S vanishes outside supp{σs}:
ω˙S (uZS ) ≤
∫
ZS
∣∣(ω˙S)u(∂S′u, ∂tu)∣∣ dS′dt
≤
∫
S(u,H)×T1
∣∣(ω˙S)u(∂S′u, ∂tu)∣∣ dS′dt
≤
∫
S(u,H)×T1
ω˙S S′ ∂S′uS′ ∂tuS′ dS′dt
≤
(
sup
Js∈J
‖ω˙S‖Js
)∫
S(u,H)×T1
∂S′uS′ ∂tuS′ dS′dt.
Using Floer’s equation, the quantity under the integral sign can be bounded by
∂S′uS′ ∂tuS′ ≤ ∂S′uS′
(
∂S′uS′ + XH(u)S′
)
≤ ∂S′u2S′ + εK‖XK‖∂S′uS′ .
50 3. SYMPLECTIC COHOMOLOGY AND DEFORMATIONS
Therefore,∫
S(u,H)×T1
∂S′uS′ ∂tuS′ dS′dt ≤
∫
S(u,H)×T1
∂S′u2S′ dS′dt+ εK‖XK‖
∫
S(u,H)×T1
∂S′uS′ dS′dt
≤ E(u) + εK‖XK‖
(∫
S(u,H)×T1
∂S′u2S′ dS′dt
) 1
2
S(u,H) 12
≤ E(u) + εK‖XK‖E(u) 12 E(u)
1
2
εH
=
(
1 +
εK
εH
‖XK‖
)
E(u).
Putting all the estimates together, we find∫
R
ω˙S(uZS ) dS ≤ µH
(
1 +
εK
εH
‖XK‖
)
E(u).
Plugging this inequality into (3.38), we finally get the desired bound for the energy
E(u) ≤ −A
ω+
H (u)
1− µH
(
1 + εKεH ‖XK‖
) . (3.39)
After this preparation, we define η by dη = τ(ω+)− τ(ω−) and claim that
ϕ({ωS},H,{JS}) : SC∗ν (Wˆ , ωˆ−, jˆ, H) −→ SC∗ν (Wˆ , ωˆ+, jˆ, H)
x 7−→
∑
v∈M(H,{JS},x′,x)
x′−x=0
(v)tη(x)−A
ω+
H (v)x′,
is welldefined and a chain map. The good definition stems from the fact that,
thanks to 3.39, for every C ∈ R, there are only finitely many connecting cylinders
v ∈ M(H, {JS}, x′, x) such that η(x) − Aω+H (v) ≤ C. Moreover, the properties a),
b), c) above implies that the boundary of a component of M(H, {JS}, y′, x) with
y′ − x = 1 is either empty or consists of two elements vxx′ ∗ ux
′
y′ and u
x
y ∗ vyy′ where
uxy ∈M(H,J−, y, x),
ux
′
y′ ∈M(H,J+, y′, x′),
vxx′ ∈M(H, {JS}, x′, x),
vyy′ ∈M(H, {JS}, y′, y).
The gluing construction in Floer theory shows also that every vxx′ ∗ux
′
y′ with the pro
perties above arises as a boundary element of a component of someM(H, {JS}, y′, x).
Therefore, to see that ϕ({ωS},H,{JS}) is a chain map, we only have to prove that
−Aω+H (ux
′
y′ ) +
(
η(x)−Aω+H (vxx′)
)
=
(
η(y)−Aω+H (vyy′)
)−Aω−H (uxy). (3.40)
First, since ω+ − ω− is νatoroidal, we have
Aω−H (uxy) = Aω+H (uxy) + (ω+ − ω−)(uxy) = Aω+H (uxy) + η(y)− η(x).
3.6. INVARIANCE UNDER DEFORMATIONS PROJECTIVELY CONSTANT ON TORI 51
Plugging this relation into (3.40), we get
−Aω+H (ux
′
y′ )−Aω+H (vxx′) = −Aω+H (vyy′)−Aω+H (uxy),
which is true since vxx′ ∗ ux
′
y′ and u
x
y ∗ vyy′ are homotopic relative ends and, as we
observed before, Aω+H is invariant under such homotopies. Thus, we have proved
that ϕ({ωS},H,{JS}) is a chain map. We denote the induced map in cohomology by[
ϕ({ωS},H,{JS})
]
.
3.6.6. Moduli spaces for smhomotopies and commutative diagrams.
In the previous subsection, we showed the existence of chain maps associated to
shomotopies. Hence, an smhomotopy (S, r) 7→ (ωS , Hr, JrS) yields a diagram
SC∗ν (W,ω−, j,H+)
ϕ
({ωS},H+,{J+S })
//
ϕ
(H+,J+− )
(H−,J−− )
SC∗ν (W,ω+, j,H+)
ϕ
(H+,J++ )
(H−,J−+ )
SC∗ν (W,ω−, j,H−)
ϕ
({ωS},H−,{J−S })
// SC∗ν (W,ω+, j,H−)
(3.41)
We claim that such diagram is commutative.
The proof follows an argument similar to the one we used to define ϕ({ωS},H,{JS}).
We subdivide the smhomotopy in rectangles [Si, Si+1]×R, so that the corresponding
constants µ
Si,Si+1
{Hr} , defined in the same way as µH , are sufficiently small. From the
commutativity of the intermediate diagrams, we infer the commutativity of the
original one.
3.6.7. Moduli spaces for sshomotopies and bijectivity of cohomology
maps. Since diagram (3.41) is commutative, the system of maps
[
ϕ({ωS},H,{JS})
]
yields a map [ϕ] : SH∗ν (W,ω−, j)→ SH∗ν (W,ω+, j). The purpose of this subsection
is to show that [ϕ] is actually an isomorphism, hence proving Proposition 3.35.
From the properties of the direct limit of groups, is enough to show that each[
ϕ({ωS},H,{JS})
]
is an isomorphism. We claim that its inverse is the map which is ob
tained by inverting the shomotopy. Namely,
[
ϕ({ωS},H,{JS})
]−1
=
[
ϕ({ω−S},H,{J−S})
]
.
First, we notice that the composition of maps in cohomology intertwines with
the concatenation of shomotopies:[
ϕ({ω−S},H,{J−S})
]
◦
[
ϕ({ω−S},H,{J−S})
]
=
[
ϕ({ωS}∗{ω−S},H,{JS}∗{J−S})
]
.
We omit the details of this fact. Thus, it is enough to prove that[
ϕ({ωS}∗{ω−S},H,{JS}∗{J−S})
]
= Id . (3.42)
Consider the standard sshomotopy that connects the concatenated shomotopy
({ωS}, H, {JS}) ∗ ({ω−S}, H, {J−S}) to the constant shomotopy (ω−, H, J−) and
notice that the chain map associated to the latter shomotopy is the identity. There
fore, to show (3.42), we need to prove that if two shomotopies are connected by an
sshomotopy, they induce the same map in cohomology. This is done by adjusting
the argument above for s and smhomotopies to sshomotopies.
The proof of Proposition 3.35 and, hence of Theorem 3.30, is completed.
CHAPTER 4
Energy levels of contact type
In this chapter we give sufficient conditions for an energy level of a magnetic
system to be of contact type and we apply the results of the previous chapter to
establish lower bounds on the number of periodic orbits, when the contact structure
is positive.
First, we observe that because of Proposition 2.6, condition (S2) on the first
Chern class is satisfied for every ν. Thus, if DM := {E ≤ 12}, it is possible to
define the groups SH∗ν (DM,ωs), whenever ω′s := ωs
∣∣
SM
is a νnondegenerate HS
of positive contact type (see Definition 2.10).
Definition 4.1. We say that a parameter s is νnondegenerate if ω′s is transver
sally νnondegenerate.
We saw that the dynamics of the magnetic flow associated to (M, g, σ) can be
studied by looking at the family of HS ω′s, where s is a real parameter. We now
want to understand when such structures are of contact type so that we can apply
techniques from Reeb dynamics and, in particular, use the abstract results proved in
the previous chapter for the computation of Symplectic Cohomology. The first step
is to determine when ω′s is exact. The answer is given by the following proposition.
Proposition 4.2. The 2form ω′s is exact if and only if one of the two alterna
tives holds
(E) either σ is exact,
(NE) or σ is not exact and M is not the 2torus.
Proof. We are going to prove the proposition by defining suitable classes of
primitives for ω′s. If σ is exact on M , we just consider the injection
Pσ −→ Pω′s
β 7−→ λs,β := λ− spi∗β. (4.1)
When σ is not exact and M 6= T2, then 2pieM[σ] σ is a curvature form for the S1
bundle pi : SM → M , since eM is the Euler number of the bundle [Kob56]. By
the discussion contained in Example 2.15 we know that pi∗σ, and hence ω′s, is exact.
More explicitly, since τ is also a connection form on SM with curvature σg, we have
the injection
Pσ−
[σ]
2pieM
σg −→ Pω′s
β 7−→ λgs,β := λ+ s
(
[σ]
2pieM
τ − pi∗β
)
(4.2)
53
54 4. ENERGY LEVELS OF CONTACT TYPE
(notice, that if M 6= T2 and σ is also exact, this formula reduces to (4.1) above).
It only remains to show that if σ is not exact and M = T2, then ω′s is not exact.
This follows from the fact that pi : ST2 → T2 is a trivial bundle, as we mentioned in
Example 2.15. More precisely, take Z : T2 → ST2 a section of this bundle. Then, Z
is a surface embedded in ST2 and∫
T2
Z∗ω′s =
∫
T2
d(Z∗λ)−
∫
T2
Z∗(spi∗σ) = −s
∫
T2
σ 6= 0.
From now on we restrict our discussion only to case (E) and (NE) contained in
the statement of Proposition 4.2. We refer to them as the exact and nonexact case.
Nonexact magnetic form on T2 will not play any role in the rest of this thesis.
Definition 4.3. Let (M, g, σ) be a magnetic system of type (E) or (NE) and
define
• Con+(g, σ) :=
{
s ∈ [0,+∞)
∣∣∣ ω′s is of positive contact type} , (4.3)
• Con−(g, σ) :=
{
s ∈ [0,+∞)
∣∣∣ ω′s is of negative contact type} . (4.4)
If we fix the manifold M and we vary the pairs (g, σ) defined on M , we get the space
of all magnetic systems of positive, respectively negative, contact type on M :
• Con+(M) :=
⋃
(g,σ)
{
(g, σ, s)  s ∈ Con+(g, σ)} ⊂ Mag(M)× [0,+∞) (4.5)
• Con−(M) :=
⋃
(g,σ)
{
(g, σ, s)  s ∈ Con−(g, σ)} ⊂ Mag(M)× [0,+∞) (4.6)
We now prove the following proposition about the sets we have just introduced.
Proposition 4.4. The sets Con+(M) and Con−(M) are open in the space
Mag(M) × [0,+∞). As a consequence, the sets Con+(g, σ) and Con−(g, σ) are
open in [0,+∞).
If s ∈ Con+(g, σ) is νnondegenerate, then SH∗ν (DM,ωs) is well defined. The
isomorphism class of SH∗ν is constant on the connected components of
a) Con+(M) ∩
(
Mage(M)× [0,+∞)
)
;
b) Con+(M), if M has positive genus;
c) Con+(S2) \
(
Mage(S
2)× [0,+∞) ∪Mag(S2)× {0}
)
.
Proof. Since a small perturbation of a contact form is still a contact form, the
first statement readily follows.
Then, observe that (DM,ωs) is convex if and only if s ∈ Con+(g, σ). Since
c1(ωs) = 0 by Proposition 2.6, its Symplectic Cohomology is well defined as soon
as s is nondegenerate. In order to prove the statement about its isomorphism
class, we apply Theorem 3.30. We only need to check that the cohomology class
[τ(ωs)] ∈ H1(LνTM,R) is projectively constant in each of the three cases presented
above. The base M is a deformation retraction of TM and under this deformation
the cohomology class [ωs] ∈ H2(TM,R) is sent to [sσ] ∈ H2(M,R). Thus, we only
need to check the analogous statement for [τ(σ)] ∈ H2(LνM,R).
4. ENERGY LEVELS OF CONTACT TYPE 55
In case a), σ is exact on M and, therefore, [τ(σ)] = 0.
In case b), we consider any νtorus u : T2 →M . Lemma 2.2 in [Mer10] implies
that u∗σ is an exact form. Therefore, [τ(σ)] = 0.
In case c), we only have the class of contractible loops. There are isomorphisms
pi1(L S2)
∼→ pi2(S2) ∼→ H2(S2,Z) ∼→ Z. Hence, H1(L S2,Z) ' Z and consequently
[τ ] : H2(S2,R) → H1(L S2,R) is an isomorphism. Consider two triples (g, σ, s)
and (g′, σ′, s′) such that s,s′ are positive and σ,σ′ are not exact. Then, [τ(ωsσ)] and
[τ(ωs′σ′)] are the same up to a positive factor if and only if the same is true for
[σ] and [σ′]. This happens if and only if the two magnetic systems lie in the same
connected component of Mag(S2) \
(
Mage(S
2)× [0,+∞) ∪Mag(S2)× {0}
)
.
In the next two subsections we are going to use the injections (4.1) and (4.2) in
order to find sufficient conditions for an energy level to be of contact type. Thanks
to Remark 2.11 and Remark 2.13, we just need to study the sign of the functions
λs,β(X
s)(x,v) = (λ− spi∗β)(X + sfV )(x,v)
= 1− βx(v)s (4.7)
for case (E) and the functions
λgs,β(X
s)(x,v) =
(
λ+
s[σ]
2pieM
τ − spi∗β
)
(X + sfV )(x,v)
= 1− βx(v)s+ [σ]f(x)
2pieM
s2 (4.8)
for case (NE).
Before performing this task, we single out some necessary conditions for ω′s to
be of contact type. Indeed, the next proposition shows that the normalised Liouville
measure is a nullhomologous invariant measure for Xs. Combining the computation
of its action (see [Pat09]) and McDuff’s criterion, contained in Proposition 2.17),
we find the mentioned necessary conditions in Corollary 4.7.
Proposition 4.5. The normalised Liouville measure χ˜ := χ2pi[µ] belongs to the
set M(Xs). Its rotation vector is the Poincare´ dual of [ω
′
s]
2pi[µ] . Therefore ρ(χ˜) = 0 if
and only if ω′s is exact. In case (E), its action is given by
Aω′s(χ˜) = 1. (4.9)
In case (NE), its action is given by
Aω′s(χ˜) = 1 + [σ]
2
2pieM [µ]
s2. (4.10)
Proof. If ζ is any 1form on SM , then by (2.24)(
ıXsζ
)
χ˜ = ζ ∧ (ıXsχ˜) = ζ ∧ ω
′
s
2pi[µ]
. (4.11)
Integrating this equality over SM , we see that the ρ(χ˜) is the Poincare´ dual of [ω
′
s]
2pi[µ] .
56 4. ENERGY LEVELS OF CONTACT TYPE
We now proceed to compute the action of χ˜. First, define I : SM → SM the
flip I(x, v) = (x,−v) and observe that I∗χ = χ. We readily see that∫
SM
βx(v)χ =
∫
SM
βx(−v)I∗χ = −
∫
SM
βx(v)χ.
Therefore,
∫
SM βx(v)χ = 0.
Consider first case (E) and let λs,β be a primitive of ω
′
s. Then,
Aω′sXs(χ˜) =
∫
SM
λs,β(X
s) χ˜ =
∫
SM
(1− βx(v)s) χ˜ = 1.
Consider now case (NE) and let λgs,β be a primitive of ω
′
s. Then,
Aω′sXs(χ˜) =
∫
SM
λgs,β(X
s) χ˜ =
∫
SM
(
1− βx(v)s+ [σ]f(x)
2pieM
s2
)
χ˜
= 1 +
[σ]
2pieM
s2
∫
SM
f(x) χ˜
= 1 +
[σ]
2pieM
s2
1
2pi[µ]
2pi
∫
M
f(x)µ
= 1 +
[σ]2
2pieM [µ]
s2.
The corollary below relates the action of χ˜ and the contact property. For the
case (NE) on surfaces of genus at least two, we first need the following definition
[Mof69, AK98, Pat09].
Definition 4.6. Let (M, g, σ) be a nonexact magnetic system on a surface of
genus at least 2. Define its helicity sh(g, σ) ∈ (0,+∞) as
sh(g, σ) :=
√
2pieM [µ]
[σ]2
(4.12)
Corollary 4.7. For exact magnetic systems or for nonexact magnetic systems
on S2, ω′s cannot be of negative contact type. Namely, Con
−(g, σ) = ∅.
For nonexact magnetic systems on a surface of genus at least 2, we have the
inclusions
• Con+(g, σ) ⊂ (0, sh(g, σ)) (4.13)
• Con−(g, σ) ⊂ (sh(g, σ),+∞). (4.14)
In particular, for s = sh(g, σ), ω
′
s is not of contact type.
4.1. Contact property for case (E)
The results of this section are classical and wellknown by the experts, except
possibly the computation of the subcritical Symplectic Cohomology on T2. Anyway,
we decided to include them here in order to put case (NE) in a wider context.
From Corollary 4.7, we know that we only need to check when λs,β is a positive
contact form. Identity (4.7) implies that
λs,β(X
s)(x,v) = 1− βx(v)s ≥ 1− βxs ≥ 1− ‖β‖s. (4.15)
4.1. CONTACT PROPERTY FOR CASE (E) 57
Therefore, λs,β(X
s) is positive provided s < ‖β‖−1. Hence, there exists a β ∈ Pσ
such that (DM,ωs, λs,β) is convex provided
s < s0(g, σ) :=
(
inf
β∈Pσ
‖β‖
)−1
.
Remark 4.8. Observe that combining [PP97, Theorem 1.1] with [CIPP98,
Theorem A], one finds that s0(g, σ) is the Man˜e´ critical value of the Abelian cover
of the Lagrangian function Lβ(x, v) := E(x, v) − βx(v) after the reparametrisation
s 7→ c(s) = 1
2s2
.
If we call [0, s1(g, σ)) the connected component of Con
+(g, σ) containing 0. The
above computation shows that s0(g, σ) < s1(g, σ). By Proposition 4.4, for all ν
nondegenerate s < s1(g, σ), SH
∗
ν (DM,ωs) ' SH∗ν (DM,ω0, j0). The latter coho
mology is known by the results of Viterbo [Vit96], SalamonWeber [SW06] and
AbbondandoloSchwarz [AS06] and we get the following statement.
Proposition 4.9. If s < s1(g, σ) is νnondegenerate, then (DM,ωs) is convex
and
SH∗ν (DM,ωs) ' H−∗(LνM,Λ). (4.16)
Remark 4.10. Observe that usually the isomorphism above is stated over Z
coefficients. However, it holds true also with Λcoefficients since Λ is torsion free
and, therefore,
H∗(LνM,Λ) ' H∗(LνM,Z)⊗Λ , SH∗ν (DM,ωs) ' SH∗ν (DM,ωs,Z)⊗Λ . (4.17)
Here we have denoted by SH∗ν (DM,ωs,Z) the direct limit of the cohomology of the
complexes (SC∗ν (DM,ωs, H,Z), δZJ ), where SC∗ν (DM,ωs, H,Z) is generated over Z
by the 1periodic orbitsof XH and δ
Z
J is defined as δJ but without using the weights
t−AωH(u). Namely,
δZJx :=
∑
u∈M(H,J,y,x)
y−x=1
(u)y,
The map δZJ is a welldefined differential, as (DM,ωs) is a Liouville domain.
The value of s1(g, σ) can be exactly estimated when M 6= T2 as the following
proposition due to Contreras, Macarini and G. Paternain shows (see [CMP04]).
Proposition 4.11. If s ≥ s0(g, σ), there exists an invariant measure ζs such
that
• pi∗ρ(ζs) = 0 ∈ H1(M,R), • ∀β ∈ Pσ, Aλs,βXs (ζs) ≤ 0. (4.18)
Therefore, if M 6= T2, Con+(g, σ) = [0, s0(g, σ)) and s1(g, σ) = s0(g, σ).
If M = T2, ω′s is not of restricted contact type for s ≥ s0(g, σ). However there
are examples for which s0(g, σ) < s1(g, σ).
Remark 4.12. It is an open question to determine whether on T2 Con+(g, σ) is
always connected, namely to see whether [0, s1(g, σ)) = Con
+(g, σ), or not.
58 4. ENERGY LEVELS OF CONTACT TYPE
In [CMP04], the existence of a magnetic system which is of contact type at
s0(g, σ) was proven using McDuff’s sufficient criterion, which we stated in Propo
sition 2.17. The drawback of this method is that the criterion is not constructive,
since it finally relies on an application of the HahnBanach Theorem. In the subsec
tion below, we outline an explicit construction of the contact form for the kind of
systems considered in [CMP04].
4.1.1. Man˜e´ Critical values of contact type on T2. The key observation is
that when M = T2, pi∗ : H1(M,R) → H1(SM,R) is not surjective. To pick a class
which is not in the image, just consider Z : T2 → ST2 a section of the S1bundle
pi : ST2 → T2. This yields the angular coordinate ϕZ : TT20 → T2pi. Therefore,
dϕZ ∈ Ω1(TT20) is a closed form which is not in the image of pi∗. If r ∈ R, we
consider the family of primitives
Pσ −→ Pω′s
β 7−→ λr,Zs,β := λ− spi∗β + rdϕZ , (4.19)
which reduces to the class introduced in (4.1) for r = 0. Namely, λ0,Zs,β = λs,β.
Recall the definition of κZ ∈ Ω1(T2) from Equation (2.9). Exploiting Formula
(2.10), we get on ST2
λr,Zs,β (X
s)(x,v) = 1− βx(v)s+ r(f(x)s− κZx (v))
≥ 1− βxs+ r(f(x)s− κZx ) . (4.20)
Now we are going to define a distinguished class of magnetic forms σ for which
the righthand side of (4.20) is positive for s = s0(g, σ).
Fix t 7→ γ(t) an embedded contractible closed curve on T2 parametrised by arc
length. Suppose that its period is Tγ and that its geodesic curvature kγ satisfies
kγ(t)− κZγ(t) ≥ ε, (4.21)
for some ε > 0. Fix a B ∈ Γ(T2) such that
(1) γ is an integral curve for B;
(2) if MB := {x ∈ T2  Bx = ‖B‖}, then MB = supp γ and ‖B‖ = 1.
Finally, let β = [B and set σ := dβ.
We claim that f(γ(t)) = kγ(t). Indeed,
f B2 = fµ(B, B) = d([B)(B, B) (∗)= B ([B(B))− B ([B(B))− ([B)([B, B])
(∗∗)
= 0− B(B2)− g(B,∇BB −∇BB)
= −B(B2) + g(∇BB, B)− 1
2
B(B2)
= B3kB − 3
2
B(B2),
where we used Cartan formula for the exterior derivative in (∗) and the symmetry
of the LeviCivita connection in (∗∗). Remember that kB is the geodesic curvature
of B defined in (2.12). The claim follows noticing that on supp γ
4.2. CONTACT PROPERTY FOR CASE (NE) 59
• B = ‖B‖ = 1;
• kB = kγ since γ is an integral line for B;
• B(B2) = 0, because the function B2 attains its global maximum there.
As a byproduct, we obtain that t 7→ (γ(t), γ˙(t) = Bγ(t)) ∈ ST2 is a periodic orbit
for X1 because γ satisfies the magnetic equation (2.16).
We claim that s0(g, σ) = ‖B‖ = 1. If we take any β′ ∈ Pσ,∫
γ
β′ =
∫
γ
β =
∫ Tγ
0
βγ(t)(γ˙(t))dt = Tγ . (4.22)
On the other hand, ∫
γ
β′ =
∫ Tγ
0
β′γ(t)(γ˙(t))dt ≤ Tγ‖β′‖, (4.23)
therefore, ‖β′‖ ≥ ‖β‖ = 1. So, the infimum in the definition of s0(g, σ) is equal to
1, it is a minimum and it is attained at β. Hence, the claim is proven.
We can now estimate from below (4.20). When s = 1,
λr,Z1,β (X
σ) ≥ (1− βx) + r(f(x)− κZx ). (4.24)
The righthand side is the sum of two pieces:
i) r(f(x)−κZx ). When r > 0, this quantity is bigger than r · ε2 on a neighbourhood
U of supp γ;
ii) 1− βx. This quantity is strictly positive outside supp γ and it vanishes on it.
In particular, there exists δ > 0 such that 1− βx ≥ δ on T2 \ U .
This means that the righthand side is positive on U and on T2 \ U as soon as
0 < r <
δ
max{0, supx/∈U κZx  − f(x)}
. (4.25)
Therefore, for r in this range, we see that (DT2, ωσ, λr,Z1,β ) is convex. We conclude
that 1 ∈ Con+(g, σ) and s1(g, σ) > s0(g, σ) = 1.
4.2. Contact property for case (NE)
To ease the notation, in this section we suppose that the magnetic form σ is
rescaled by a constant factor, in such a way that [σ] = 2pieM . This operation will
only induce a corresponding rescaling of the parameter s and, hence, will not affect
our study. We begin with two easy examples and then we move to the general
discussion subdivided in the subsections below.
Example 4.13. Denote by g0 the metric of curvature Kg0  = 1 on M and
consider the magnetic systems (M, g0, µg0). We see that µg0 − [µg0 ]2pieM σg0 = 0. Hence,
we can choose β = 0 and the corresponding family of primitives λg0s,0.
If M = S2, then (4.8) yields λg0s,0(X
s) = 1 + s2. Thus, Con+(g0, µg0) = [0,+∞).
If M is a surface of genus at least 2, (4.8) reduces to λg0s,0(X
s) = 1 − s2. Thus,
Con+(g0, µg0) = [0, 1) and Con
−(g0, µg0) = (1,+∞). Notice that in this case,
sh(g0, µg0) = 1 is the only value of the parameter which is not of contact type.
60 4. ENERGY LEVELS OF CONTACT TYPE
Example 4.14. Consider a convex twosphere. Namely, we endow S2 with a
metric g of positive curvature. Take the magnetic system (S2, g, σg = Kµ). As in
the example above, we can choose β = 0 and the family of primitives λgs,0. In this
case, (4.8) reduces to λgs,0(X
s) = 1 +Ks2 > 0. Thus, Con+(g, σg) = [0,+∞). This
is one of the hints to Conjecture E.
4.2.1. High energy levels. Let us start by checking when s ∈ Con+(g, σ).
Identity (4.8) implies that
λgs,β(X
s)(x,v) = 1− βx(v)s+ f(x)s2 ≥ 1− ‖β‖s+ inf fs2 (4.26)
Consider the quadratic equation 1 − ‖β‖s + inf fs2 = 0. If inf f ≤ 0, we have only
one positive root. If inf f > 0 we either have two positive roots or two nonreal roots.
In any case, call s−(g, σ, β) the smallest positive root and set s−(g, σ, β) = +∞ if
the equation does not have real roots. Observe that 1 − ‖β‖s + inf fs2 > 0 for
s ∈ [0, s−(g, σ, β)), no matter the sign of inf f . We write
s−(g, σ) := sup
β∈Pσ−σg
s−(g, σ, β). (4.27)
Notice that s−(g, σ) is the smallest positive root (with the same convention as before
if there are no real roots) of the quadratic equation 1−m(g, σ)s+inf fs2 = 0, where
m(g, σ) := inf
β∈Pσ−σg
‖β‖. (4.28)
Thus, we conclude that there exists s−(g, σ) ≥ s−(g, σ) such that [0, s−(g, σ)) is
the connected component of Con+(g, σ) containing 0.
Proposition 4.4 implies the following corollary for surfaces of high genus.
Corollary 4.15. If M is a surface of genus at least 2 and s < s−(g, σ) is
νnondegenerate, then (DM,ωs) is convex and
SH∗ν (DM,ωs) ' SH∗ν (DM,ω0) ' H−∗(LνM,Λ). (4.29)
We now deal with the case of S2. If we fix s < s−(g, σ), there exists β ∈ Pσ−σg
such that s < s−(g, σ, β). Consider the Riemannian metric g0 with Kg0 = 1 and
let gr := rg + (1 − r)g0 be the linear interpolation between g and g0. Define a
corresponding family of βr ∈ Pσ−σgr such that β1 = β. By compactness of the
interval,
sˆ−(g, σ, β) := inf
r∈[0,1]
s−(gr, σ, βr)
is positive.
Take s∗ ≤ s such that 0 < s∗ < sˆ−(g, σ, β) and consider the deformation
s′ ∈ [s∗, s], s′ 7→ (DS2, ωs′σ, λgs′,β).
Since s′ ≤ s, the boundary stays convex for every s′. Moreover, the deformation is
projectively constant because s′ > 0. Then, define a second deformation
r ∈ [0, 1], r 7→ (DgrS2, ωgrs∗σ, λgrs∗,βr),
4.2. CONTACT PROPERTY FOR CASE (NE) 61
where we have made explicit the dependence of the disc bundle and the symplectic
form on the metric using the superscript gr. Since s∗ < sˆ−(g, σ, β), the boundary
remains convex also in this case. Moreover, the cohomology class of ωgrs∗σ does not
change.
Finally, consider a last deformation with a parameter u ∈ [0, 1]. Define the
1parameter family of 2forms σu = uσ + (1 − u)σg0 = σg0 + udβ0 and take the
corresponding family
u 7→ (Dg0S2, ωg0s∗σu , λg0s∗,uβ0).
Let fg0u be the function such that σu = f
g0
u µg0 . Then, f
g0
u = uf
g0
1 + (1− u)Kg0 and
inf fg0u = u inf fg0 + (1 − u)Kg0 ≥ inf fg01 (observe that fg0u and Kg0 have the same
integral over M). Therefore, s∗ < sˆ−(g, σ, β) ≤ s−(g0, σ, β0) ≤ s−(g0, σu, uβ0), for
every u ∈ [0, 1]. Hence, the boundary stays convex also during this last deformation
and the cohomology class of ωg0s∗σu does not change since [s∗σu] is fixed.
Summing up, we have shown that for s < s−(g, σ), (g, σ, s) and (g0, σg0 , s∗) lie in
the same connected component of Con+(S2)\
(
Mage(S
2)×[0,+∞)∪Mag(S2)×{0}
)
,
for any s∗ > 0 (remember that in Example 4.13, we proved that s−(g0, σg0) = +∞).
We can now apply Proposition 4.4.
Corollary 4.16. If M = S2 and 0 < s < s−(g, σ) is nondegenerate, then
(DS2, ωs) is convex and
SH∗(DgS2, ωgsσ) ' SH∗(Dg0S2, ωg0s∗σg0 ), (4.30)
where g0 is the metric with constant curvature 1 and s∗ is any positive number.
In the next section, we are going to compute the righthand side of (4.30). In the
remainder of this section we are going to discuss other two cases where we can use
Identity (4.8) to find energy levels of contact type. The original idea is contained in
[Pat09, Remark 2.2].
4.2.2. Low energy levels on S2. Let us go back to Inequality (4.26) and see
what happens when the polynomial 1 − s‖β‖ + s2 inf f has a second positive root
s++(g, σ, β). We observed that this happens if and only if inf f > 0. Because of the
normalisation we made at the beginning of this section, we see that this can happen
only if σ is a symplectic form on S2. Under these hypotheses, we have that the
righthand side of (4.26) is positive for s ∈ s++(g, σ, β). As before we denote
s++(g, σ) = inf
β∈Pσ−σg
s++(g, σ, β), (4.31)
which can also be defined as the biggest positive root of 1−m(g, σ)s+ inf fs2 = 0.
We conclude that there exists s++(g, σ) ≤ s++(g, σ) such that (s++(g, σ),+∞) is
the unbounded connected component of Con+(g, σ).
Corollary 4.17. Let M = S2 and σ be a symplectic form. If s > s++(g, σ) is
nondegenerate, then (DS2, ωs) is convex and
SH∗(DgS2, ωgsσ) ' SH∗(Dg0S2, ωg0s∗σg0 ), (4.32)
62 4. ENERGY LEVELS OF CONTACT TYPE
where g0 is the metric with constant curvature 1 and s∗ is any positive number.
Again, we refer to the next section for the computation of the righthand side.
4.2.3. Low energy levels on a surface of high genus. We now want to
investigate levels of negative contact type. Thanks to Corollary 4.7, they can arise
only on a surface of genus at least 2. Let us bound λgs,β(X
s) from above using (4.8):
λgs,β(X
s)(x,v) = 1− βx(v)s+ f(x)s2 ≤ 1 + ‖β‖s+ sup fs2.
If sup f ≥ 0 the righthand side cannot be less than zero. If inf f < 0, there exists
a unique positive root of the associated quadratic equation s−+(g, σ, β) such that the
righthand side is less than zero, for every s > s−+(g, σ, β). Denote
s−+(g, σ) := inf
β∈Pσ−σg
s−+(g, σ, β), (4.33)
which can also be defined as the unique positive root of 1 +m(g, σ)s− inf fs2 = 0.
Therefore, we conclude that there exists s−+(g, σ) ≤ s−+(g, σ) such that (s−+(g, σ),+∞)
is the unbounded connected component of Con−(g, σ).
We cannot compute Symplectic Cohomology for compact concave symplectic
manifolds (namely, for symplectic manifolds whose boundary is of negative contact
type). A possible solution to this problem would be to consider only invariants
of the boundary such as (embedded) contact homology. Another possibility would
be to look for a compact convex symplectic manifold (W,ω) that could be used
to cap off (DM,ωs) from outside in order to form a closed symplectic manifold
(DM unionsqSM W,ωs unionsqSM ω). We plan to deal with this second approach in a future
research project.
Remark 4.18. Thanks to Corollary 4.7, we have the chain of inequalities
0 < s−(g, σ) ≤ sh(g, σ) ≤ s−+(g, σ). (4.34)
Moreover, G. Paternain proved in [Pat09] that
0 < sc(g, σ) ≤ sh(g, σ), (4.35)
where
sc(g, σ) = inf
θ˜∈P σ˜
‖θ˜‖g˜
is the critical value of the universal cover after the reparametrisation c(s) = 1
2s2
. It
is an open problem to study the relation between s−(g, σ) and sc(g, σ).
4.3. Symplectic Cohomology of a round sphere
In this subsection we compute the Symplectic Cohomology of (DS2, ωsσg0 ), for
s > 0, when g0 is the metric of constant curvature 1.
First, we look for a primitive λˆg0s,0 of ωsσg0 on the whole TS
2
0 which extends
λg0s,0 = λ + sτ ∈ Ω1(SS2). Using Identity (2.6), we readily find λˆg0s,0 = λ + s τ2E .
Integrating the Liouville vector field associated to λˆg0s,0 starting from SS
2 yields a
4.3. SYMPLECTIC COHOMOLOGY OF A ROUND SPHERE 63
convex neighbourhood jg0s,0 = (rs, ps) : TS
2
0 ↪→ R × SS2 such that SS2 = {rs = 0}.
Let us differentiate E with respect to this coordinate:
dE
drs
= dE(∂rs) = ωsσg0 (∂rs , X
sσg0
E ) = λˆ
g0
s,0(X
sσg0
E ) = 2E + s
2.
Dividing both sides by 2E + s2 and integrating between 0 and rs, we get
rs = log
√
2E + s2
1 + s2
, (4.36)
or, using the auxiliary variable, Rs = e
rs ,
Rs =
√
2E + s2
1 + s2
.
From this equation we see two things. First, that Rs (or equivalently rs) is also
smooth at the zero section and, hence, it can be extended to a smooth function on the
whole tangent bundle. Second, that the image of jg0s,0 is
(
log
(
s√
1+s2
)
,+∞
)
× SS2.
In particular, the flow of ∂rs ∈ Γ(TS20) is positively complete and, therefore, jg0s,0 is
actually a cylindrical end. Thus, (TS2, ωsσg0 , λˆ
g0
s,0) is the symplectic completion of
(DS2, ωsσg0 , λ
g0
s,0).
Let us look now at the dynamics on SS2. We claim that all orbits are periodic
and the prime orbits have all the same minimal period Ts =
2pi√
1+s2
. The claim can
be proven either by finding explicitly all the curves with constant geodesic curvature
(these are the boundaries of geodesic balls) or by using Gray Stability Theorem. The
latter strategy yields an isotopy Fs′ : SS
2 → SS2, with s′ ∈ [0, s], such that
F ∗s′(λ+ s
′τ) =
1√
1 + (s′)2
λ.
Since the Reeb vector field corresponding to λ is X, whose orbits are all periodic
and with minimal period 2pi, the claim follows.
We are ready to compute the Symplectic Cohomology. Take as a family of
increasing Hamiltonian, linear at infinity, the functions Hks := k
√
1 + s2Rs, with
k ∈ N. The associated Hamiltonian flow generates an S1action on TS2 with period
2pik. Hence, the only 1periodic orbits of the flow are the constant orbits, which lie on
the zero section. Let us compute their ConleyZehnder index (for the computation
in the Lagrangian setting we refer to [FMP13, Lemma 5.4]).
The linearisation of the flow dΦ
Hks
t : T(x,0)TS
2 → T(x,0)TS2 can be described
as follows. First, we choose standard coordinates on TS2, close to the point (x, 0).
Then, we compute the differential of the vector field XHks at (x, 0) using these
coordinates. We find the matrix
d(x,0)XHks = k
(
0 Id
0 sx
)
Since d(x,0)Φ
Hks
t = exp(t ·dx0XHks ) (here exp denote the exponential of a linear endo
morphism), the eigenvalues of the linearisation are 1 and eikst, both with algebraic
multiplicity 2. Observe that since 1 is an eigenvalue, (x, 0) is degenerate (actually,
64 4. ENERGY LEVELS OF CONTACT TYPE
the zero section form a MorseBott component of critical points since the eigenvalue
in the normal directions is eikst, which is different from 1 when t is positive). For
this reason, we take the lower semicontinuous extension of the index and find
µlCZ(x, 0) = 2
⌊
ks
2pi
⌋
+ 1 + (−1) = 2
⌊
ks
2pi
⌋
,
where we used the additivity of the ConleyZehnder index under direct products.
Consider now a timedependent compact perturbation Hks,δk of H
k
s such that
all the 1periodic orbits of the new system are nondegenerate. Here δk > 0 is a
perturbative parameter that we take arbitrarily small depending on k. We claim that
the direct limit for k → +∞ of the symplectic cohomology groups with Hamiltonian
Hks,δk is zero. Indeed, let γ be a 1periodic orbit of the Hamiltonian system associated
to Hks,δk . Since δk is small, γ is close to a constant solution on the zero section. By
the lower semicontinuity of the index, [γ] = 2 − µCZ(γ) ≤ 2 − 2
⌊
ks
2pi
⌋
. Therefore,
the symplectic cohomology with Hamiltonian Hks,δk is zero in degree bigger than
2 − 2 ⌊ ks2pi⌋. Since 2 − 2 ⌊ ks2pi⌋ → −∞ as k → +∞, the direct limit is zero in every
degree. Thus, we have proven the following proposition.
Proposition 4.19. For every s > 0, there holds SH∗(DS2, ωsσg0 ) = 0.
Corollary 4.20. If (S2, g, σ) is a nonexact magnetic system and s is a non
degenerate parameter such that s < sˆ−(g, σ) or s > sˆ++(g, σ), then
SH∗(DS2, ωs) = 0. (4.37)
4.4. Lower bound on the number of periodic orbits
The computation of Symplectic Cohomology we performed in the previous sec
tions can be used to prove the existence of periodic orbits on SM .
Proposition 4.21. Let (S2, g, σ) be an exact magnetic system. If s < s0(g, σ),
then Xs ∈ Γ(SS2) has at least one periodic orbit. If the iterates of such orbit are
transversally nondegenerate, then there is another geometrically distinct periodic
orbit on SS2.
Proof. From [Zil77], we know the singular homology with Zcoefficients of the
free loop space of S2. It is zero in negative degree and, for k ∈ N, we have
Hk(L S
2,Z) =
Z if k = 0 or k is odd,Z⊕ Z/2Z if k is even and positive. (4.38)
Since H−∗(L S2,Z) 6= H∗+2(S2,Z), we have H−∗(L S2,Λ) 6= H∗+2(S2,Λ) by
Remark 4.10. Therefore there exists at least one prime periodic orbit γ by Corollary
3.21. Call T its period.
Suppose that γ and all its iterates are transversally nondegenerate. Construct
inductively a sequence k 7→ Hk ∈ Hˆ′0 (see Definition 3.18) on the completion D̂S2
of the disc bundle, so that
• THk < THk+1 ;
4.4. LOWER BOUND ON THE NUMBER OF PERIODIC ORBITS 65
• kT < THk ;
• Hk = Hk+1 on D̂S2ck , where kT = (hk)′(eck) = (hk+1)′(eck).
Take small perturbations Hkε ∈ Hˆ0 and let Jk be a Hkε admissible almost complex
structure. Define SC∗(k) := SC∗0 (DS2, ωs, Hkε ), let δk be the Floer differentials
δk : SC
∗(k) → SC∗(k), let SH∗(k) be the associated cohomology groups and let
ϕk : SC
∗(k) → SC∗(k) be the continuation maps. Denote by SH∗(∞) the direct
limit of SH∗(k). For every k˜ = 1, . . . , k, we have two nonconstant generators
of SC∗(k): γk˜min and γ
k˜
Max. They have degree 2 − µCZ(γk˜) and 2 − µCZ(γk˜) − 1,
respectively. By Corollary 3.20, we know that ϕk is the inclusion map.
Since SH∗(∞) is nonzero in arbitrarily low degree, µCZ(γk) > 0 for every k ≥ 1,
by the properties of the index. We distinguish three cases.
a) µCZ(γ) ≥ 2. If this happens, then for every k˜, µCZ(γk˜+1)− µCZ(γk˜) > 1. Hence,
for ∗ ≤ −2, there can be at most one generator for SC∗(k). This contradicts the
fact that SH−4(∞) ' Λ⊕ Λ/2Λ.
b) µCZ(γ) = 1 and γ is hyperbolic. If k > 1, SC
1(k) is generated by γmin and
SC0(k) generated by γMax, γ
2
min and p, where p is the image under the map
C2(S2,Λ)→ SC2(k) of a generating cochain in C2(S2,Λ) ' Λ. We claim that δk
is zero in degree 1 and 0. This would imply that the class of γmin is nontrivial in
SH1(k). Together with the fact that ϕk is the inclusion, we get a contradiction
to the fact that SH1(∞) = 0.
Since there are no generators in degree 2, it is clear that δk is zero in degree 1.
Since γ is a good orbit, δkγMax = 0 and since p is the image of a cochain under a
chain map, we also have δkp = 0. Finally, we compute 0 = δ
2
kγ
2
Max = δk(2t
aγ2min),
which implies that δkγ
2
min = 0 as well.
c) µCZ(γ) = 1 and γ is elliptic. Under these hypotheses, there exists a maximum
k0 ≥ 1 such that µCZ(γk0) = 1. If k ≥ k0, SC1(k) is generated by the elements
γk˜min, with k˜ ∈ [1, k0] and SC0(k) is generated by p and the elements γk˜Max, with
k˜ ∈ [1, k0]. As the action AωsHkε is decreasing with the period, we know that γ
k0
min
has action smaller than γk˜Max, for k˜ < k0. By Corollary 3.19, < δkγ
k˜
Max, γ
k0
min >= 0.
Moreover, δkp = 0 and, since γ
k0 is a good orbit, also < δkγ
k0
Max, γ
k0
min >= 0. We
conclude that γk0min is a cochain, which is not a coboundary in SC
1(k). Together
with the fact that ϕk is the inclusion, we get a contradiction to the fact that
SH1(∞) = 0.
Since each case leads to a contradiction, we have proved that there exists a periodic
orbit geometrically distinct from γ.
Proposition 4.22. Let (T2, g, σ) be an exact magnetic system. If s < s1(g, σ),
then Xs ∈ Γ(ST2) has at least one periodic orbit in every nontrivial free homotopy
class. If such periodic orbit is transversally nondegenerate, then there is another
periodic orbit in the same class.
Proof. If there are no periodic orbits SH∗ν (DT2, ωs) = 0. If we only have one
periodic orbit and such orbit is non degenerate, then rkSH∗ν (DT2, ωs) ∈ {0, 2}.
66 4. ENERGY LEVELS OF CONTACT TYPE
In any case rkSH∗ν (DT2, ωs) ≤ 2. On the other hand, we claim that LνT2 is
homotopy equivalent to T2. If γν is a reference loop in the class ν, the two homotopy
equivalences are given by
T2 −→ LνT2
x 7−→ γν,x(t) := γν(t) + x;
LνT2 −→ T2
γ 7−→ γ(0).
Thanks to Proposition 4.9, this leads to the contradiction
2 ≥ rkSH∗ν (DT2, ωs) = rkH−∗(LνT2,Λ) = rkH−∗(T2,Λ) = 4.
Proposition 4.23. Let (M, g, σ) be a magnetic system on a surface of genus at
least two. If σ is exact and s < s0(g, σ) or σ is not exact and s < sˆ−(g, σ), then
Xs ∈ Γ(SM) has at least one periodic orbit in every nontrivial free homotopy class.
Proof. If there are no orbits in the class ν 6= 0, SH∗ν (DM,ωs) = 0. However,
LνM is homotopy equivalent to S1, if ν is nontrivial. Thanks to Proposition 4.9
or Corollary 4.15, this leads to the contradiction
0 = SH∗ν (DM,ωs) = H−∗(LνM,Λ) = H−∗(S
1,Λ) 6= 0.
Proposition 4.24. Let (S2, g, σ) be a nonexact magnetic system. If the para
meter s belongs to (0, s−(g, σ))∪ (s++(g, σ),+∞), then Xs ∈ Γ(SS2) has at least one
periodic orbit. If the iterates of such periodic orbit are transversally nondegenerate,
then there is another geometrically distinct periodic orbit on SS2.
Proof. The proof is similar to the one of Proposition 4.21 and, therefore, we
only discuss the parts where we have to argue differently from there.
Since SH∗(DS2, ωs) = 0 6= H∗+2(S2,Λ), there exists at least one prime periodic
orbit γ by Corollary 3.27. Call T its period.
Suppose that γ and all its iterates are transversally nondegenerate. Construct
the sequences Hk and Hkε as in Proposition 4.21 and let J
k
ε be a sequence in J (Hk)
(see Definition 3.22). Define SC∗(k), δk, SH∗(k), ϕk and SH∗(∞) as before.
For every k˜ = 1, . . . , k, we have two nonconstant generators of SC∗(k): γk˜min
and γk˜Max. They have degree 2 − µCZ(γk˜) and 2 − µCZ(γk˜) − 1, respectively. By
Corollary 3.26, we know that ϕk is the inclusion map.
If µCZ(γ
k) ≤ 0 for every k ≥ 1, then the degrees of the nonconstant periodic
orbits are all positive. By Corollary 3.27, this means that SH−2(∞) ' H0(S2,Λ),
which is a contradiction. Therefore, we assume that µCZ(γ
k) > 0, for every k ≥ 1.
We distinguish four cases.
a) µCZ(γ) ≥ 3. We claim that there exists k0 such that µCZ(γk0+1)− µCZ(γk0) > 2
and γk0 is good. This implies that γk0Max represents a nonzero class on SC
∗(k),
for k ≥ k0. Taking the direct limit, we conclude that SH∗(∞) 6= 0, contradicting
our assumption. Thus, we only need to prove the claim.
We use the iteration formula for the index (see [Lon02]):
µCZ(γ
k) =
2bkϑc+ 1, ϑ ∈ (0,+∞) \Q, if γ is elliptic;kµCZ(γ), if γ is hyperbolic.
4.4. LOWER BOUND ON THE NUMBER OF PERIODIC ORBITS 67
If γ is hyperbolic, then µCZ(γ
k+1) − µCZ(γk) = µCZ(γ) > 2 and the claim is
satisfied by any good iterate of γ. If γ is elliptic, then ϑ > 1 and
µCZ(γ
k+1)− µCZ(γk) = 2 + 2 (b(k + 1)(ϑ− 1)c − bk(ϑ− 1)c) .
Therefore, there exists k0 such that b(k0 + 1)(ϑ− 1)c− bk0(ϑ− 1)c ≥ 1. For such
k0 we have µCZ(γ
k0+1)− µCZ(γk0) > 2. Since all the iterates of an elliptic orbit
are good, the claim is proven also in this case.
b) µCZ(γ) = 2. In this case γmin has degree zero and SC
∗(k) = 0 for ∗ > 0.
Therefore, δkγmin = 0. Moreover, SC
−1(k) is generated only by γMax and since
γ is a good orbit, we know that < δkγMax, γmin >= 0. This implies that γmin
represents a nonzero class in SH0(k). Taking the direct limit, we conclude that
SH0(∞) 6= 0, contradicting our assumption.
c) µCZ(γ) = 1 and γ is hyperbolic. This case leads to a contradiction following the
same argument given in Proposition 4.21.
d) µCZ(γ) = 1 and γ is elliptic. Also this case leads to a contradiction following the
same argument given in Proposition 4.21, but we have to use the filtration with
the period (see Section 3.4.2), instead of the filtration with the action.
Since each case leads to a contradiction, we have proved that there exists a periodic
orbit geometrically distinct from γ.
Remark 4.25. For the case of S2, we do not know whether the orbits we find
are homotopic to a fibre or not.
CHAPTER 5
Rotationally symmetric magnetic systems on S2
In this chapter we aim at studying the contact property when (S2γ , gγ) is a surface
of revolution with profile function γ and σ = µγ is the area form. We are going to
estimate the interval [s−(gγ , µγ), s++(gγ , µγ)] in terms of geometric properties of γ and
try to understand, at least with numerical methods, the gaps [s−(gγ , µγ), s−(gγ , µγ)]
and [s++(gγ , µγ), s
+
+(gγ , µγ)]. To ease the notation we are going to use the parameter
m instead of s in this chapter. Hence, we define
Xmγ := mX
mµγ
Eγ
∣∣
SS2γ
, ωm,γ := mωµγ
m
,
m+,γ :=
1
s−(gγ , µγ)
, m−,γ :=
1
s++(gγ , µγ)
,
m+,γ :=
1
s−(gγ , µγ)
, m−,γ :=
1
s++(gγ , µγ)
,
Conγ :=
{
m ∈ (0,+∞)
∣∣∣ 1
m
∈ Con+(gγ , µγ)
}
∪ {0} .
5.1. The geometry of a surface of revolution
To construct a surface of revolution, take a function γ : [0, `γ ]→ R and consider
the conditions:
1) γ(t) = 0 if t = 0 or t = `γ and is positive otherwise,
2) γ˙(0) = 1, γ˙(`γ) = −1 and γ˙(t) < 1 for t ∈ (0, `γ),
3) all even derivatives of γ vanish for t ∈ {0, `γ},
4) the following equality is satisfied∫ `γ
0
γ(t) dt = 2. (5.1)
Definition 5.1. A function γ satisfying the first three hypotheses of the list is
called a profile function. If also the fourth one holds, we say that γ is normalised.
Let S2γ be the quotient of [0, `γ ] × T2pi with respect to the equivalence relation
that collapses the set {0} × T2pi to a point and the set {`γ} × T2pi to another point.
We call these points the south and the north pole. Outside the poles the smooth
structure is given by the coordinates (t, θ) ∈ (0, `γ) × T2pi, which also determine a
welldefined orientation on S2γ .
On S2γ we put the Riemannian metric gγ , defined in the (t, θ) coordinates by the
formula gγ = dt
2 + γ(t)2dθ2. This metric extends smoothly to the poles because of
conditions 2) and 3) listed before. Moreover, condition 4) yields the normalisation
volgγ (S
2
γ) = 4pi.
69
70 5. ROTATIONALLY SYMMETRIC MAGNETIC SYSTEMS ON S2
Let us denote by (t, θ, vt, vθ) the associated coordinates on the tangent bundle. If
ϕ := ϕ∂t : S
(
S2γ \{south pole, north pole}
)→ T2pi is the angular function associated
to the section ∂t, we have the relations
vt = cosϕ, vθ =
sinϕ
γ(t)
. (5.2)
As a consequence, (t, ϕ, θ) are coordinates on SS2γ , which are compatible with the
orientation OSS2 defined in Section 2.2. Using the coframe (λ, τ, η), we can express
the coordinate frame (∂˜t, ∂ϕ, ∂˜θ) in terms of the frame (X,V,H) and vice versa:
∂˜t = cosϕX − sinϕH
∂ϕ = V
∂˜θ = γ sinϕX + γ cosϕH + γ˙V.
(5.3)
X = cosϕ∂˜t − γ˙ sinϕ
γ
∂ϕ +
sinϕ
γ
∂˜θ
V = ∂ϕ
H = − sinϕ∂˜t − γ˙ cosϕ
γ
∂ϕ +
cosϕ
γ
∂˜θ,
(5.4)
We have put a tilde on ∂˜t and ∂˜θ to distinguish them from the coordinate vectors ∂t
and ∂θ associated to the coordinates (t, θ) on S
2
γ .
As anticipated at the beginning of this chapter, we consider as the magnetic
form on the surface SS2γ , the Riemannian area form µγ . This is a symplectic form
which satisfies the normalisation [µγ ] = 4pi introduced in Section 4.2. In coordinates
(t, θ) we have µγ = γdt ∧ dθ. With this choice f ≡ 1 and µγ − σg = (1−K)µγ . We
recall that the Gaussian curvature for surfaces of revolution is K = − γ¨γ .
If we define
mγ := m(gγ , µγ) = inf
β∈P(1−K)µγ
‖β‖,
we have the formula
m±,γ =
mγ ±
√
m2γ − 4
2
.
In particular, the length of the interval [m−,γ ,m+,γ ] is equal to
√
m2γ − 4, a monotone
increasing function of mγ . This observation readily gives a sufficient condition for
having the contact property at every energy level.
Corollary 5.2. If mγ < 2, then Conγ = [0,+∞).
In the next section we compute mγ , showing that the infimum in its definition
is actually a minimum. Namely, there exists βγ ∈ P(1−K)µγ such that mγ = ‖βγ‖.
5.2. Estimating the set of energy levels of contact type
Consider an arbitrary closed Riemannian manifold (M, g). Let Z ∈ Γ(M) be a
vector field that generates a 2piperiodic flow of isometries on M . The projection
operator on the space of Zinvariant kforms ΠkZ : Ω
k(M)→ ΩkZ(M) is defined as
∀τ ∈ Ωk(M), ΠkZ(τ) :=
1
2pi
∫ 2pi
0
(ΦZθ′)
∗τ dθ′. (5.5)
5.2. ESTIMATING THE SET OF ENERGY LEVELS OF CONTACT TYPE 71
Proposition 5.3. The operators ΠkZ commute with exterior differentiation:
d ◦ΠkZ = Πk+1Z ◦ d. (5.6)
The projection Π1Z does not increase the norm ‖ · ‖ defined in (2.2):
∀β ∈ Ω1(M), ‖Π1Z(β)‖ ≤ ‖β‖. (5.7)
Proof. For the proof of the first part we refer to [Boo86, Section: The De
Rham groups of Lie groups]. For the latter statement we use that ΦZ is a flow of
isometries. We take some (x, v) ∈ TM and compute∣∣∣Π1Z(β)x(v)∣∣∣ = ∣∣∣∣ 12pi
∫ 2pi
0
βΦZ
θ′ (x)
(
dxΦ
Z
θ′v
)
dθ′
∣∣∣∣
≤ 1
2pi
∫ 2pi
0
‖β‖∣∣dxΦZθ′v∣∣ dθ′
=
1
2pi
∫ 2pi
0
‖β‖v dθ′
= ‖β‖v.
Thus, Π1Z(β)x ≤ ‖β‖. The proposition follows by taking the supremum over x.
Let us apply this general result to S2γ . Consider the coordinate vector field ∂θ.
This extends to a smooth vector field also at the poles and Φ∂θ is a 2piperiodic
flow of isometries on the surface. Applying Proposition 5.3 to this case, we get the
following corollary.
Corollary 5.4. The 2form (1 − K)µγ is ∂θinvariant. Hence, Π1∂θ sends
P(1−K)µγ into itself.
Proof. The first statement is true because Φ∂θ is a flow of isometries and thus
µγ and K are invariant under the flow. To prove the second statement, we observe
that, if β ∈ P(1−K)µγ , the previous proposition yields
d
(
Π1∂θ(β)
)
= Π2∂θ(dβ) = Π
2
∂θ
((1−K)µγ) = (1−K)µγ .
Hence, Π1∂θ(β) ∈ P(1−K)µγ .
Suppose now that β lies in P(1−K)µγ ∩ Ω1∂θ(S2γ). Thus, β = βt(t)dt + βθ(t)dθ.
The function βθ = β(∂θ) is uniquely defined by dβ = (1−K)µγ and the fact that it
vanishes on the boundary of [0, `γ ]. Indeed,
dβ = d
(
βtdt+ βθdθ
)
= β˙θdt ∧ dθ = β˙θ
γ
µγ (5.8)
Recalling the formula for K, we have β˙θ = γ + γ¨. Hence, βθ = Γ + γ˙, where
Γ : [0, `γ ]→ [−1, 1] is the only primitive of γ such that Γ(0) = −1. Notice that
1) Γ is increasing,
2) Γ(`γ) = −1 +
∫ `γ
0 γ(t) dt = −1 + 2 = 1,
3) the odd derivatives of Γ vanish at {0, `γ}.
72 5. ROTATIONALLY SYMMETRIC MAGNETIC SYSTEMS ON S2
Since βθ and its derivatives of odd orders are zero for t = 0 and t = `γ , the 1form
βγ := βθdθ is well defined also at the poles and belongs to P(1−K)µγ ∩ Ω1∂θ(S2γ).
Finally, the norm of this new primitive is less than or equal to the norm of β:
∣∣β(t,θ)∣∣ =
√
β2t +
β2θ
γ2
≥
∣∣∣∣βθγ
∣∣∣∣ = ∣∣βγ(t,θ)∣∣. (5.9)
Summing up, we have proven the following proposition.
Proposition 5.5. There exists a unique βγ ∈ P(1−K)µγ that can be written in
the form βγ = βγθ (t)dθ. It satisfies mγ = ‖βγ‖. Moreover βγθ = Γ + γ˙ and
‖βγ‖ = sup
t∈[0,`γ ]
∣∣∣∣Γ(t) + γ˙(t)γ(t)
∣∣∣∣ . (5.10)
Thus, we see that we can compute mγ directly from the function γ. As an
application, we now produce a simple case where mγ can be bounded from above.
Proposition 5.6. Suppose γ : [0, `γ ]→ R is a normalised profile function such
that γ(t) = γ(`γ − t). If K is increasing in the variable t, for t ∈ [0, `γ/2], then
mγ ≤ 1. Therefore, Conγ = [0,+∞).
Proof. We observe that the functions Γ, γ˙ and, hence, βγθ are odd with respect
to t = `γ/2. This means, for example, that β
γ
θ (t) = −βγθ (`γ − t). Therefore, in order
to compute mγ , we can restrict the attention to the interval [0, `γ/2]. We know
that βγθ (0) = β
γ
θ (`γ/2) = 0 and we claim that, if K is increasing, β
γ
θ is positive in
the interior. Indeed, observe that {t ∈ [0, `γ/2]  β˙γθ = 0} is an interval and that
β˙γθ is positive before this interval and it is negative after. Therefore, either β
γ
θ is
constantly zero or it does not have a local minimum in the interior. Thus, it cannot
assume negative values.
Let us estimate βγθ /γ at an interior absolute maximiser t0. The condition
d
dt
∣∣
t=t0
βγθ
γ = 0 is equivalent to(
Γ(t0) + γ˙(t0)
)
γ˙(t0)
γ2(t0)
= 1−K(t0). (5.11)
Since Γ(t0)+ γ˙(t0) = β
γ
θ (t0) ≥ 0 and γ˙(t0) > 0, we see that 1−K(t0) ≥ 0. Moreover,
using that Γ(t0) < 0, we get (
γ˙(t0)
γ(t0)
)2
> 1−K(t0). (5.12)
Finally, exploiting Equation (5.11) again, we find
mγ =
βγθ (t0)
γ(t0)
=
(
1−K(t0)
)γ(t0)
γ˙(t0)
≤
√
1−K(t0) ≤ 1. (5.13)
The fact that Conγ = [0,+∞) now follows from Corollary 5.2.
To complement the previous proposition we show that if, on the contrary, we
assume that the curvature of S2γ is sufficiently concentrated at one of the poles,
mγ can be arbitrarily large. We are going to prove this behaviour in the class of
5.2. ESTIMATING THE SET OF ENERGY LEVELS OF CONTACT TYPE 73
convex surfaces since in this case we also have a strategy, explained in Section 5.4,
to compute numerically the action of invariant measures. Using McDuff’s criterion,
this will enable us to estimate the gaps [m−,γ ,m−,γ ] and [m+,γ ,m+,γ ].
Before we need a preliminary lemma. Recall that S2γ is convex, i.e. K ≥ 0, if
and only if γ¨ ≤ 0.
Lemma 5.7. For every 0 < δ < pi2 and for every ε > 0, there exists a normalised
profile function γδ,ε such that S
2
γδ,ε
is convex and
γ˙δ,ε(δ) < ε. (5.14)
Proof. Given δ and ε, we find a ∈ (2δpi , 1) such that 0 < cos
(
δ
a
)
< ε. This is
achieved by taking δa very close to pi/2 from below. Consider the profile function
γa : [−pia2 , pia2 ]→ [0, a] of a round sphere of radius a, where the domain is taken to be
symmetric to zero to ease the following notation. It is defined by γa(t) := a cos(
t
a).
Then, γ˙a(−pia2 +δ) = cos
(
δ
a
)
< ε and so, up to shifting the domain again, Inequality
(5.14) is satisfied. However, γa is not normalised since∫ pia
2
−pia
2
γa(t)dt = 2a
2 < 2.
In order to get the normalisation in such a way that Inequality (5.14) is not spoiled,
we are going to stretch the sphere in the interval
(−(pia2 − δ), pia2 − δ).
We claim that, for every C > 0 there exists a diffeomorphism FC : R→ R with
the property that
• it is odd: ∀t ∈ R, FC(t) = −FC(−t);
• for t ≥ pia2 − δ, FC(t) = t+ C and for t ≤ −(pia2 − δ), FC(t) = t− C;
• F˙C ≥ 1;
• for t ≤ 0, F¨C(t) ≥ 0 and for t ≥ 0, F¨C(t) ≤ 0.
Such a map can be constructed as a time C flow map ΦψC , where ψ : R → R is an
odd increasing function such that, for t ≥ pia2 − δ, ψ(t) = 1.
Consider the function γCa : [−C − pia2 , C + pia2 ]→ R, where γCa (s) := γa(F−1C (s)).
One readily check that γCa (up to a shift of the domain) is a profile function satisfying
the convexity conditions and for which (5.14) holds. To finish the proof it is enough
to find a positive real number C2 such that
∫
R γ
C2
a = 2. Since we know that γ
0
a = γa
and
∫
R γa < 2, it suffices to show that
lim
C→+∞
∫
R
γCa (s)ds = +∞.
Observe that b := γCa (−C − pia2 + δ) = γa(−pia2 + δ) = a sin( δa) > 0. Then, for
s ∈ [−C − pia2 + δ, C + pia2 − δ], γCa (s) ≥ b and we have the lower bound∫
R
γCa (s)ds ≥
∫ C+pia
2
−δ
−C−pia
2
+δ
γCa (s)ds ≥
∫ C+pia
2
−δ
−C−pia
2
+δ
b ds = 2(C +
pia
2
− δ)b.
The last quantity tends to infinity as C tends to infinity.
74 5. ROTATIONALLY SYMMETRIC MAGNETIC SYSTEMS ON S2
Proposition 5.8. For every C > 0, there exists a convex surface with total area
4pi such that mγ > C.
Proof. Fix an ε0 < 1. Take any δ <
√
1− ε0 and consider the normalised
profile function γδ,ε0 given by the lemma. We know that
γδ,ε0(δ) =
∫ δ
0
γ˙δ,ε0(t)dt ≤
∫ δ
0
1 dt = δ.
In the same way we find Γδ,ε0(δ) ≤ −1 + δ2. From this last inequality we get
Γδ,ε0(δ) + γ˙δ,ε0(δ) < −1 + δ2 + ε0 < 0.
This yields the following lower bound for mγδ,ε0 :
mγδ,ε0 ≥
∣∣∣∣Γδ,ε0(δ) + γ˙δ,ε0(δ)γδ,ε0(δ)
∣∣∣∣ ≥ 1− ε0δ − δ.
The proposition is proven taking δ small enough.
To sum up, we saw that the rotational symmetry gives us a good understanding
of the set [0,m−,γ) ∪ (m+,γ ,+∞). Understanding the set [0,m−,γ) ∪ (m−,γ ,+∞),
or even better Conγ , is more subtle. In Section 5.4 we perform this task only
numerically and when the magnetic curvature Km = m
2K + 1 is positive.
As a first step, in the next section we will briefly study the symplectic reduction
associated to the symmetry and the associated reduced dynamics (for the general
theory of symplectic reduction we refer to [AM78]). Proposition 5.10 and the
numerical computation outlined in Section 5.4 suggest that, if Km > 0, the contact
property holds. In particular, if K ≥ 0, every energy level should be of contact
type. To complete the picture, we show in Proposition 5.11 that the assumption on
the magnetic curvature is not necessary and in Proposition 5.12 that there are cases
where the magnetic curvature is not positive and that are not of contact type.
5.3. The symplectic reduction
Observe that the flow Φ∂θ on S2γ lifts to a flow dΦ
∂θ on SS2γ . Since Φ
∂θ is
a flow of isometries, dΦ∂θθ′ in coordinates is simply translation in the variable θ:
dΦ∂θθ′ (t, ϕ, θ) = (t, ϕ, θ+θ
′). Hence, dΦ∂θ is generated by ∂˜θ. As the flow Φ∂˜θ = dΦ∂θ
is 2piperiodic and acts freely on SS2γ , we can take its quotient ŜS
2
γ with respect to
this T2piaction. Furthermore, the quotient map ̂ : SS2γ → ŜS2γ is a submersion.
The variables t and ϕ descend to coordinates defined on ŜS2γ minus two points, which
are the fibres of the unit tangent bundle over the south and north pole. In these
coordinates we simply have ̂(t, ϕ, θ) = (t, ϕ). In particular, ŜS2γ is diffeomorphic
to a 2sphere.
Any τ ∈ Ωk
∂˜θ
(SS2γ) such that ı∂˜θ
τ = 0 passes to the quotient and yields a well
defined form on ŜS2γ . The 2form ı∂˜θ
νγ falls into this class and, hence, there exists
Θγ ∈ Ω2(ŜS2γ) such that ı∂˜θνγ = ̂∗Θγ . Moreover, the form Θγ is symplectic on
ŜS2γ . On the other hand, X
m
γ is also ∂˜θinvariant thanks to Equation (5.4). So there
5.3. THE SYMPLECTIC REDUCTION 75
exists X̂mγ ∈ Γ(ŜS2γ) such that d̂(Xmγ ) = X̂mγ . We claim that this new vector field
is ΘγHamiltonian. We start by noticing that if β
γ is as defined in Proposition 5.5
and λγm := mλ − pi∗βγ + τ , then λγm ∈ Ω1widetilde∂θ(SS2γ). Then, Cartan identity
implies
ıXmγ (ı∂˜θ
νγ) = −ı∂˜θ(ıXmγ νγ) = −ı∂˜θωm,γ = −ı∂˜θ(dλ
γ
m) = −L∂˜θλ
γ
m + d
(
ı
∂˜θ
λγm
)
= d
(
λγm(∂˜θ)
)
. (5.15)
Define Im,γ := λ
γ
m(∂˜θ). Since Im,γ is ∂˜θinvariant, there exists Îm,γ : ŜS2γ → R such
that Im,γ = ̂∗Îm,γ . Thus, reducing equality (5.15) to ŜS2γ , we find that X̂mγ is the
ΘγHamiltonian vector field generated by −Îm,γ . Using Equation (5.3), we find the
coordinate expression
Îm,γ(t, ϕ) = mγ(t) sinϕ− Γ(t). (5.16)
As a byproduct, we also observe that Im,γ is an integral of motion for X
m
γ .
Let us consider now the two auxiliary functions Î±m,γ : [0, `γ ] → R defined by
Î±m,γ(t) := Îm,γ(t,±pi/2) = ±mγ(t)− Γ(t). We know that
Î−m,γ(t) ≤ Îm,γ(t, ϕ) ≤ Î+m,γ(t), (5.17)
with equalities if and only if ϕ = ±pi/2. On the one hand, we have Î+m,γ ≥ −1 and
Î+m,γ(t) = −1 if and only if t = `γ . On the other hand, Î+m,γ attains its maximum in
the interior. Indeed,
d
dt
Î+m,γ = mγ˙ − γ. (5.18)
and so ddt Î
+
m,γ(0) = m > 0. Since Î
+
m,γ(0) = 1, the maximum is also strictly bigger
than 1. A similar argument tells us that the maximum of Î−m,γ is 1 and it is attained
at 0 and the minimum of Î−m,γ is strictly less than −1 and it is attained in (0, `γ).
As a consequence, max Îm,γ = max Î
+
m,γ > 1 and min Îm,γ = min Î
−
m,γ < −1.
In the next proposition we deal with the critical points Ĉritm,γ of Îm,γ and study
their relations with the critical points Ĉrit
±
m,γ of Î
±
m,γ . We are going to show that, if
Km > 0, the only elements in Ĉritm,γ are the (unique) maximiser and the (unique)
minimiser. In this case the dynamics of X̂mγ is very simple: besides the two rest
points, all the other orbits are periodic and wind once in the complement of these
two points.
Proposition 5.9. There holds
Ĉritm,γ = Ĉrit
−
m,γ ×
{
−pi
2
}⋃
Ĉrit
+
m,γ ×
{
+
pi
2
}
.
Moreover, t0 ∈ Ĉrit±m,γ if and only if ±mγ˙(t0) = γ(t0) and {(t0,±pi/2, θ)} is the
support of a closed orbit for Xmγ . All the periodic orbits, whose projection to S
2
γ is
a latitude, arise in this way. Every regular level of Îm,γ is the support of a closed
orbit of X̂mγ and its preimage in SS
2
γ is an X
m
γ invariant torus.
Finally, if Km > 0, Ĉrit
±
m,γ contains only the absolute maximiser (respectively
minimiser) of Î±m,γ. We denote this unique element by t±m,γ.
76 5. ROTATIONALLY SYMMETRIC MAGNETIC SYSTEMS ON S2
tHI0L t+HI0L {HΓL
t
1
I0
1
I
I
`
m,Γ
+
I
`
m,Γ

0
Figure 1. Graphs of the functions Î−m,γ and Î+m,γ
Proof. The first statement follows from the fact that ∂ϕÎm,γ = 0 if and only if
ϕ = ±pi/2. Recalling (5.18) we see that ±mγ˙(t) = γ(t) is exactly the equation for
the critical points of Î±m,γ . The statement about the relation between closed orbits
of Xmγ and latitudes follows from the fact that at a critical point of Îm,γ , X̂
m
γ = 0.
Hence, on its preimage Xmγ is parallel to ∂˜θ. By the implicit function theorem the
regular level sets of Îm,γ and Im,γ are closed submanifolds of codimension 1. In the
latter case they are tori since Xmγ is tangent to them and nowhere vanishing.
We prove now uniqueness under the hypothesis on the curvature. We carry out
the computations for Î+m,γ only. To prove that the absolute maximiser is the only
critical point, we show that if t0 is critical, the function is concave at t0. Indeed,
d2
dt2
Î+m,γ(t0) = mγ¨(t0)− γ˙(t0) = mγ(t0)
(
γ¨(t0)
γ(t0)
− γ˙(t0)
mγ(t0)
)
= −mγ(t0)
(
K(t0) +
1
m2
)
< 0.
The picture above shows qualitatively Î−m,γ and Î+m,γ when Km is positive.
In order to decide whether ω′m,γ is of contact type or not, the first thing to do
is to compute the action of latitudes.
Proposition 5.10. Take t0 ∈ (0, `γ) such that γ˙(t0) 6= 0 and let mt0 :=
∣∣∣γ(t0)γ˙(t0) ∣∣∣.
The lift of the latitude curve {t = t0} parametrised by arc length and oriented by(
sign γ˙(t0)
)
∂θ is the support of a periodic orbit for X
mt0
γ . We call ζt0 the associated
invariant probability measure. Its action is given by
Aω
′
mt0
,γ
X
mt0
γ
(ζt0) =
γ(t0)
2 − γ˙(t0)Γ(t0)
γ˙(t0)2
(5.19)
and Imt0 ,γ
∣∣
supp ζt0
= γ˙(t0)A(ζt0). As a result, if Kmt0 > 0, then A(ζt0) > 0.
5.3. THE SYMPLECTIC REDUCTION 77
Proof. The curve u 7→
(
t0, sign γ˙(t0)
pi
2 ,
m sign γ˙(t0)
γ(t0)
u
)
is a periodic orbit by the
previous proposition. On the support of this orbit we have
mt0vθ =
∣∣∣∣γ(t0)γ˙(t0)
∣∣∣∣ sign γ˙(t0)γ(t0) = 1γ˙(t0) (5.20)
and, as a consequence,
λγmt0
(X
mt0
γ ) =
γ(t0)
2
γ˙(t0)2
− βγθ (t0)
1
γ˙(t0)
+ 1 =
γ(t0)
2 − γ˙(t0)Γ(t0)
γ˙(t0)2
. (5.21)
Since this is a constant, we get Identity (5.19) for the action.
The second identity is proved using the definition of Imt0 ,γ :
Imt0 ,γ
∣∣
supp ζt0
=
∣∣∣∣γ(t0)γ˙(t0)
∣∣∣∣ γ(t0) sign γ˙(t0)− Γ(t0) = γ˙(t0)γ(t0)2 − γ˙(t0)Γ(t0)γ˙(t0)2 .
Under the curvature assumption, Imt0 ,γ is maximised or minimised at supp ζt0 ac
cording to the sign of γ˙(t0). In both cases Imt0 ,γ
∣∣
supp ζt0
and γ˙(t0) have the same
sign. Hence, also the third statement is proved.
This proposition shows that, when Km > 0, the action of the periodic orbits that
project to latitudes is not an obstruction for ω′m,γ to be of contact type. Thus, as we
also discuss in the next subsection, one could conjecture that under this hypothesis
ω′m,γ is of contact type. On the other hand, we claim that having Km > 0 is
not a necessary condition for the contact property to hold. For this purpose it is
enough to exhibit a nonconvex surface for which mγ < 2. This can be achieved as
a consequence of the fact that the curvature depends on the second derivative of γ,
whereas mγ depends only on the first derivative.
We can start from S2γ0 , the round sphere of radius 1, and find a nonconvex
surface of revolution S2γ , which is C
1close to the sphere and coincides with it around
the poles. This implies that mγ = supt∈[0,`γ ]
∣∣∣Γ(t)+γ˙(t)γ(t) ∣∣∣ can be taken smaller than 2
since it is close as we like to mγ0 = 0. Hence, every energy level of (S
2
γ , gγ , µγ) is of
contact type and the following proposition is proved.
Proposition 5.11. The condition Km > 0 is not necessary for Σm to be of
contact type.
On the other hand, we now show that is not true, in general, that the contact
property holds on every energy level.
Proposition 5.12. There exists a symplectic magnetic system (S2, g, σ) that
has an energy level not of contact type.
Proof. We will achieve this goal by finding m and γ such that Xmγ has a closed
orbit projecting to a latitude with negative action. Then, the proof is complete
applying Proposition 2.17.
Fix some ε ∈ (0, 1). We claim that, for every δ ∈ (0, pi/2), there exists a
normalised profile function γδ,ε such that
γ˙δ,ε(δ) < −ε. (5.22)
78 5. ROTATIONALLY SYMMETRIC MAGNETIC SYSTEMS ON S2
Such profile function can be obtained as in Lemma 5.7. Take an a ∈
(
δ
pi ,
δ
pi/2
)
(this
is equivalent to δ ∈ (pia2 , pia)) such that − sin ( δa − pi2 ) < −ε. Consider the profile
function γa : [−pia/2, pia/2] → R of a round sphere of radius a. Thanks to the last
inequality, γ˙a(δ − pia2 ) < −ε and, therefore, γa satisfies (5.22) (up to a shift of the
domain). Now we stretch an interval compactly supported in (δ − pia2 , pia/2) by a
family of diffeomorphisms FC as we did in Lemma 5.7 (though here the condition
on the second derivative of FC is not necessary). In this way we obtain a family of
profile functions γCa satisfying (5.22). Since the area diverges with C, we find C2 > 0
such that γC2a is normalised. This finishes the proof of the claim.
Since
∣∣γ˙δ,ε(δ)∣∣ ≤ 1, we have that γδ,ε(δ) ≤ δ and Γδ,ε(δ) ≤ −1 + δ2. The latitude
at height t = δ of such surface is a closed orbit for X
mδ,ε
γδ,ε , where mδ,ε :=
γδ,ε(δ)
γ˙δ,ε(δ) .
Using formula (5.19) we see that the action of the corresponding invariant measure
ζδ,ε is negative for δ small enough:
A(ζδ,ε) ≤ δ
2 − (−ε)(−1 + δ2)
ε2
= −1
ε
+ o(δ) < 0.
5.4. Action of ergodic measures
When Km > 0, we also have a way to compute numerically the action of ergodic
invariant measures. We consider only ergodic measures since they are the extremal
points of the set of probability invariant measures by Choquet’s Theorem and, there
fore, it is enough to check the positivity of the action of these measures, in order
to apply Proposition 2.17. Every ergodic measure ζ is concentrated on a unique
level set {Im,γ = I(ζ)}, for some I(ζ) ∈ R. Moreover, if I(ζ) = I(ζ ′) there exists
a rotation Φ∂˜θθ′ such that
(
Φ∂˜θθ′
)
∗ζ = ζ
′. Since the action is ∂˜θinvariant, we deduce
that it is a function of I(ζ) only and we can define A : [min Im,γ ,max Im,γ ]→ R. We
already have an expression for the action at the minimum and maximum of Im,γ .
We now give a formula for the action when I ∈ (min Im,γ ,max Im,γ).
Every integral line z : R→ SS2 of Xmγ , such that Im,γ(z) = I oscillates between
the latitudes at height t−(I) and t+(I). Their numerical values can be easily read
off from the graphs of Î±m,γ drawn in Figure 1. If we take z with t(z(0)) = t−(I),
there exists a smallest u(I) > 0 such that t
(
z(u(I))
)
= t+(I). By Birkhoff’s ergodic
theorem
A(I) = 1
u(I)
∫ u(I)
0
(
m2 −mβθ(t) sinϕ
γ(t)
(z(u)) + 1
)
du. (5.23)
Using this identity, we computed with Mathematica the action when S2γ is an ellip
soid. We found that is positive on every energy level in the interval [m−,γ ,m+,γ ]. By
Proposition 2.17, this shows numerically that Conγ = [0,+∞), hence corroborating
the conjecture that Σm is of contact type if Km > 0. On the other hand, we know
that, when the ellipsoid is very thin, its curvature is concentrated on its poles and
hence, by Proposition 5.8, the set [m−,γ ,m+,γ ] is not empty. Therefore, these data
would also show numerically that the inclusion [0,m−,γ) ∪ (m+,γ ,+∞) ⊂ Conγ can
be strict.
CHAPTER 6
Dynamically convex Hamiltonian structures
In this chapter we introduce dynamically convex Hamiltonian structures, for
which we have sharper results on the set of periodic orbits. They were introduced
by Hofer, Wysocki and Zehnder in [HWZ98] as a way of generalising the contact
structures arising on the boundary of convex domains in C2 (see Example 6.4 below)
with a notion which makes sense in the contact category. Later on, this notion has
been extended by Hryniewicz, Licata and Saloma˜o [HLS13] to lens spaces and, at
the moment this thesis is being written, Abreu and Macarini [AM] are developing a
general theory of dynamical convexity, which will eventually embrace also nonexact
magnetic systems on surfaces of higher genus (see Section 7.4.1).
The abstract results we present below will play a crucial role in Chapter 7, when
we will analyse symplectic magnetic systems on S2.
Definition 6.1. Let ω be a HS of contact type on L(p, q). We say that ω is
dynamically convex if, for every contractible periodic orbit γ, µlCZ(γ) ≥ 3. By abuse
of terminology, we will call a contact primitive of ω dynamically convex, as well.
Remark 6.2. Thanks to (2.32) the condition on the index is equivalent to
I(ΨC,Υγ ) ⊂ (1,+∞) (see (2.34)).
Remark 6.3. If pi : S3 → L(p, q) is the quotient map, then ω is dynamically
convex if and only if pi∗ω is dynamically convex.
Example 6.4. If Σ ↪→ C2 is a closed hypersurface bounding a convex domain,
then λst
∣∣
Σ
is dynamically convex.
Example 6.5. We remarked in Example 2.14 that, if (M, g) is a Riemannian
surface, λ
∣∣
SM
is a contact form. Let M = S2 and notice that SM ' L(2, 1). Harris
and G. Paternain showed in [HP08] that, if 14 ≤ K < 1, then λ
∣∣
SS2
is dynamically
convex.
For further examples of dynamical convexity, in the context of the circular planar
restricted threebody problem and of the rotating Kepler problem, we refer the
reader to [AFF+12] and [AFFvK13], respectively.
As we show in the next two sections, dynamical convexity has two main conse
quences for the associated Reeb flow: the existence of a symplectic Poincare´ section
and the existence of an elliptic periodic orbit.
6.1. Poincare´ sections
The main result on the dynamics of dynamically convex Hamiltonian Structures
reads as follows.
79
80 6. DYNAMICALLY CONVEX HAMILTONIAN STRUCTURES
Theorem 6.6 ([HWZ98],[HLS13]). Let ω be a HS on L(p, q) and suppose,
furthermore, that, if p > 1, ω is nondegenerate. If ω is dynamically convex, then
there exists a Poincare´ section of disctype D2 → L(p, q) (which is a psheeted cover
on ∂D2) for the characteristic distribution of ω.
Remark 6.7. The case of S3, namely p = 1, is due to Hofer, Wysocki and
Zehnder, while the case p > 1 is due to Hryniewicz, Licata and Saloma˜o. It is
likely that the nondegeneracy condition for p > 1 can be dropped by repeating the
lengthy argument contained in [HWZ98, Section 6–8]. However, for the applications
to periodic orbits of a degenerate system we can simply apply Theorem 6.6 after we
have lifted the problem from L(p, q) to S3. The details of such argument will be
explained below.
For the convenience of the reader we now recall the notion of Poincare´ section.
Definition 6.8. Let N be a closed 3manifold. A Poincare´ section for Z ∈ Γ(N)
is a compact surface i : S → N such that
• it is an embedding on the interior S˚ := S \ ∂S;
• i(∂S) is the disjoint union of a finite collection of embedded loops {γk} and
i
∣∣
i−1(γk)
: i−1(γk)→ γk is a finite cover;
• the vector field Z is transverse to i(S˚) and every γk is the support of a
periodic orbit for Z;
• every flow line of Z hits the surface in forward and backward time.
If z ∈ S˚, let t(z) be the smallest positive number such that FS˚(z) := ΦZt(z)(z) ∈ S˚.
The map FS˚ : S˚ → S˚ defines a diffeomorphism called the Poincare´ first return map.
Finally, if Z is a positive section for kerω, where ω is a HS on N , a map i
satisfying the requirements above is called a Poincare´ section for ω.
If Z and N are as in Definition 6.8, the discrete dynamical system (S˚, FS˚) carries
important information about the qualitative dynamics on N , since periodic points
of FS˚ correspond to the periodic orbits of Z different from the γk’s.
When Z is a positive section of kerω, with ω ∈ Ω2(N) a Hamiltonian Structure,
(S˚, i∗ω) is a symplectic manifold with finite area and FS˚ is a symplectomorphism.
If we suppose, in addition, that S is a disc, then N ' L(p, q) (see [HLS13])
and Proposition 5.4 in [HWZ98] implies that FS˚ is C
0conjugated to a homeomor
phism of the disc preserving the standard Lebesgue measure. The work of Brouwer
([Bro12]) and Franks ([Fra92] and [Fra96]) on areapreserving homeomorphisms
of the disc imply that either FS˚ has only a single periodic point, which is a fixed
point for FS˚ , or there are infinitely many periodic points.
Corollary 6.9. Suppose ω ∈ Ω2(L(p, q)) is a HS having a Poincare´ section of
disctype. Then, kerω has 2 periodic orbits γ1 and γ2 whose lifts to S
3, γ̂1 and γ̂2,
form a Hopf link (namely they are unknotted and  lk(γ̂, γ̂′) = 1). Either these are
the only periodic orbits of kerω or there are infinitely many of them. The second
alternative holds if there exists a periodic orbit whose lift to S3 is knotted or if there
are two periodic orbits γ and γ′ such that  lk(γ̂, γ̂′) 6= 1.
6.1. POINCARE´ SECTIONS 81
Example 6.10. Let us verify the above general results in a concrete example.
For any pair (p, q) of positive real numbers, consider the ellipsoid
Σp,q :=
{
(z1, z2) ∈ C2
∣∣∣ z12
2p
+
z22
2q
= 1
}
⊂ C2.
Since Σp,q is convex, we know by Example 6.4 that it is also dynamically convex
and, hence, by Theorem 6.6, its Reeb flow has a Poincare´ section of disctype. We
can construct explicitly one such disc as follows. First, using the Hamilton equation
for the Hamiltonian (z1, z2) 7→ z1
2
p +
z22
q , we get R
λ =
∂θ1
p +
∂θ2
q , where θi is the
angular coordinate in the ziplane. Thus, Φ
Rλ
t (z1, z2) = (e
it
p z1, e
it
q z2) and we see
that, for any θ ∈ T2pi, Sθ := {(z1, z2) ∈ Σp,q
∣∣ θ1 = θ} is a Poincare´ section of disc
type parametrised by z2. The return map is FS˚θ(z2) = e
2pii p
q z2, namely a rotation
by the angle 2pi pq . Such rotation has the origin as the unique fixed point if and only
if pq is irrational. When
p
q is rational, every orbit is periodic.
If ω is dynamically convex and, when p > 1, it is also assumed to be non
degenerate, Theorem 6.6 directly implies that the hypotheses of Corollary 6.9 are
satisfied and, thus, we have a dichotomy between two and infinitely many periodic
orbits for kerω. We now aim to prove that the dichotomy still holds for a dynamically
convex HS, without any nondegeneracy assumption.
We restrict to p = 2, since SS2 ' L(2, 1) and we will need only this case in the
applications.
Regard L(2, 1) as the quotient of S3 by the antipodal map A : S3 → S3. The
quotient map pi : S3 → L(2, 1) is a double cover with A as the only nontrivial deck
transformation. There is a bijection Z 7→ Ẑ between Γ(L(2, 1)) and ΓA(S3) ⊂ Γ(S3)
the subset of Ainvariant vector fields. The antipodal map permutes the flow lines
of Ẑ. Moreover, a lift of a trajectory for Z is a trajectory for Ẑ and the projection
of a trajectory for Ẑ is a trajectory for Z. In the next lemma we restrict this
correspondence to prime contractible periodic orbits of Z. We are grateful to Marco
Golla for communicating to us the elegant proof about the parity of the linking
number presented below.
Lemma 6.11. There is a bijection between contractible prime orbits z of Z and
pairs of antipodal prime orbits {ẑ, A(ẑ)} of Ẑ such that ẑ and A(ẑ) are disjoint.
Furthermore, the linking number lk(ẑ, A(ẑ)) between them is even.
Proof. Associate to a contractible periodic orbit z its two distinct lifts ẑ1 and
ẑ2 = A(ẑ1). Since z is contractible both lifts are closed. They are also prime since
a lift of an embedded path is still embedded. Suppose that the two lifts intersect.
This implies that there exist points t1 and t2 such that ẑ1(t1) = ẑ2(t2). Applying pi
to this equality, we find z(t1) = z(t2) and so t1 = t2 modulo the period of z. Hence,
ẑ1 = ẑ2 contradicting the fact that the two lifts are distinct.
For the inverse correspondence, associate to two antipodal disjoint prime periodic
orbits {ẑ, A(ẑ)} their common projection pi(ẑ). The projected curve is contractible
since its lifts are closed. Furthermore, it is also prime since, if pi(ẑ)(t1) = pi(ẑ)(t2),
82 6. DYNAMICALLY CONVEX HAMILTONIAN STRUCTURES
either ẑ(t1) = A(ẑ)(t2) and ẑ ∩ A(ẑ) 6= ∅ or ẑ(t1) = ẑ(t2) and t1 = t2 modulo the
period of ẑ.
We now compute the linking number between the two knots. Consider S3 as
the boundary of B4 the unit ball inside R4 and denote still by A the antipodal
map on B4, which extends the antipodal map on S3. Take an embedded surface
S1 ⊂ B4 such that ∂S1 = ẑ1 and transverse to the boundary of B4. By a small
perturbation we can also assume that 0 ∈ B4 does not belong to the surface. The
antipodal surface S2 := A(S1) has the curve ẑ2 as boundary and lk(ẑ1, ẑ2) is equal
to the intersection number between S1 and S2. By perturbing again S1 we can
suppose that all the intersections are transverse. Indeed, if we change S1 close to
a point z of intersection, this will affect S2 = A(S1) only near the antipodal point
A(z) = −z, which is different from z since the origin does not belong to S1. Now that
transversality is achieved, we claim that the number of intersections is even. This
stems from the fact that, if z ∈ S1 ∩ S2, then A(z) ∈ A(S1) ∩ A(S2) = S2 ∩ S1 and
z and A(z) are different since z 6= 0. As a consequence, the algebraic intersection
number between the two surfaces is even as well and the lemma follows.
Remark 6.12. In the proof of the lemma, the sign of the intersection between
S1 and S2 at z is the same as the sign at A(z), since A preserves the orientation.
Thus, we cannot conclude that the total intersection number is zero. Indeed, for
any k ∈ Z one can find a pair of antipodal knots, whose linking number is 2k.
Proposition 6.13. If ω is a dynamically convex HS on L(2, 1), then kerω has
either two or infinitely many periodic orbits. In the first case, the two orbits are
noncontractible.
Proof. By Corollary 6.9 there exist two prime closed orbits ẑ1 and ẑ2 of kerpi
∗ω
forming a Hopf link and if there is any other periodic orbit geometrically distinct
from these two, kerpi∗ω has infinitely many periodic orbits.
We claim that z1 := pi(ẑ1) and z2 := pi(ẑ2) are geometrically distinct closed orbits
for kerω on L(2, 1). If, by contradiction, z1 coincides with z2, by Lemma 6.11, ẑ1
and ẑ2 are antipodal and their linking number is even. This is a contradiction since
lk(ẑ1, ẑ2) = 1. Therefore, we conclude that z1 and z2 are distinct. On the other
hand, if kerpi∗ω has infinitely many periodic orbits the same is true for kerω. Hence,
also kerω has either 2 or infinitely many distinct periodic orbits.
If kerω has a prime contractible periodic orbit w, its lifts ŵ1 and ŵ2 are disjoint,
antipodal and prime periodic orbits for kerpi∗ω by Lemma 6.11. Since lk(ŵ1, ŵ2) is
even, {ŵ1, ŵ2} 6= {ẑ1, ẑ2} and, therefore, there are at least three distinct periodic
orbits for kerpi∗ω. So there are infinitely many periodic orbits for kerpi∗ω and,
hence, also for kerω.
6.2. Elliptic periodic orbits
Generally speaking, once we have established the existence of a periodic orbit
(γ, T ) for a flow ΦZ , we can use it to understand the behaviour of the dynamical
6.2. ELLIPTIC PERIODIC ORBITS 83
system in a neighbourhood of γ by looking at the spectral properties of t 7→ dγ(0)ΦZt ,
the linearisation of the flow at the periodic orbit.
We have seen in Definition 2.10 that when γ is a nondegenerate periodic orbit
of a Hamiltonian Structure ω, we have only two possibilities: either the transverse
spectrum lies on the real line (γ hyperbolic) or it lies on the unit circle of the complex
plane (γ elliptic).
As a sample of the applications one might get in the first case we mention the
work of Hofer, Wysocki and Zehnder in [HWZ03]. The authors proved in Theorem
1.9 that a generic Reeb flow Rτ on S3 without Poincare´ sections of disctype, has
a hyperbolic orbit whose stable and unstable manifolds intersect in a homoclinic
orbit. Following [Mos65, Chapter III], this yields the existence of a Bernoulli shift
embedded in the flow of Rτ through the construction of local Poincare´ sections.
Since a Bernoulli shift has infinitely many periodic points, ΦR
τ
has infinitely many
periodic orbits as well.
In contrast to the chaotic behaviour caused by the presence of the Bernoulli
shift, when γ is elliptic the flow is expected to be stable and quasiperiodic close
to the periodic orbit. As explained in [Mos65, Section 2.4.d], if the transverse
spectrum does not contain any root of unity, the KAM theorem implies the existence
of a fundamental system of open neighbourhoods of the orbit, whose boundaries are
invariant under the flow. In particular, each of these neighbourhoods is Rτ invariant.
Hence, γ is a stable periodic orbit and, as a result, the Reeb flow is nonergodic.
Existence of elliptic periodic orbits for the Reeb flow on the boundary of convex
domains in C2 has been proved in particular cases and it is an open problem raised
by Ekeland [Eke90, page 198] to determine whether an elliptic orbit is to be found
on every system of this kind. Its existence has been showed if
• the curvature satisfies a suitable pinching condition [Eke86];
• the hypersurface is symmetric with respect to the origin [DDE95].
The latter case is the most interesting to us since, in view of the double cover
S3 → SS2, lifted contact forms will automatically be invariant with respect to the
antipodal map.
Recently, Abreu and Macarini have announced an extension of the results con
tained in [DDE95] to dynamically convex symmetric systems.
Theorem 6.14 ([AM]). If ω is a dynamically convex Hamiltonian Structure on
L(p, q), with p > 1, then it has an elliptic periodic orbit.
In the same work, the authors will also define a notion of dynamical convexity
for contact forms not necessarily on lens spaces. This has potential applications to
symplectic magnetic fields on surfaces of genus at least two on low energy levels. We
will comment on such generalisation in Section 7.4.1.
CHAPTER 7
Low energy levels of symplectic magnetic flows on S2
In this chapter we study in more detail low energy levels of magnetic flows
(S2, g, σ) when σ is a symplectic form normalised in such a way that [σ] = 4pi.
Throughout this chapter we are going to use the parameter m and the associated
notation introduced in Section 2.3 just before Remark 2.8.
Remark 7.1. Consider more generally a mechanical system (S2, g, σ, U) with σ
symplectic. We saw in Section 2.3.1 that when m is small and ‖U‖ is small compared
to m2, the dynamics on Σm is the dynamics of a corresponding magnetic system on
low energy level. Thus, we can apply to this case the results presented below.
Moreover, we saw in Section 2.3.2 a concrete example of mechanical system
(S2, gˆ, kmσgˆ, Uˆk) obtained by reduction from (SO(3), g, 0, U). The associated mag
netic system is (S2, gˆm,Uˆk ,
k
mσgˆ), with ‖gˆm,Uˆk − gˆ‖ ≤
2‖Uˆk‖
m2
and
2‖Uˆk‖
m2
=
2‖U + k2
2ν2
‖
m2
≤ 2‖U‖
m2
+
k2
m2
∥∥∥∥ 1ν2
∥∥∥∥ .
Thus, we are allowed to approximate the reduced dynamics of (SO(3), g, 0, U) with
momentum k and energy m
2
2 , with a symplectic magnetic system on a low energy
level provided ‖U‖ and k2 are small with respect to m2.
7.1. Contact forms on low energy levels
Given β ∈ Pσ−σg , fix for the rest of the chapter a primitive λgm,β = mλ−pi∗β+τ
of ω′m. We know that there exists mβ > 0 such that, for m ∈ [0,mβ), the function
hm,β(x, v) := λ
g
m,β(X
m)(x,v) = m
2 − βx(v)m+ f(x) (7.1)
is positive and, as a consequence λgm,β is a positive contact form. Denote by R
m,β :=
Rλ
g
m,β , the Reeb vector field, by Φm,β its flow and by ξm,β := kerλgm,β the contact
distribution. We readily compute
Rm,β =
Xm
hm,β
=
m
hm,β
X +
f
hm,β
V. (7.2)
We proceed to relate the time parameter of a flow line for Φm,β and the length of
its projection on S2.
Lemma 7.2. If γ : [0, T ]→ SS2 is a flow line of Φm,β, then
m
maxhm,β
T ≤ `(pi(γ)) ≤ m
minhm,β
T, (7.3)
where ` denotes the length of a curve in S2 with respect to the metric g.
85
86 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Proof. We compute the norm of the tangent vector of pi(γ):∣∣∣∣ ddtpi(γ)
∣∣∣∣ = ∣∣∣∣dγpi( ddtγ
)∣∣∣∣ = ∣∣∣∣ mhm,β γ
∣∣∣∣ = mhm,β . (7.4)
Plugging this identity into the formula for the length of pi(γ), we get
`(pi(γ)) =
∫ T
0
∣∣∣∣ ddtpi(γ)
∣∣∣∣ dt = ∫ T
0
m
hm,β
dt (7.5)
and the lemma follows bounding the integrand from below and from above.
We define a global ω′msymplectic trivialisation of ξm,β in the next lemma.
Lemma 7.3. The contact structure ξm,β admits a global ω′msymplectic frame:
Hˇm,β :=
H + βx(xv)V√
hm,β
Xˇm,β :=
X +
(
βx(v)−m
)
V√
hm,β
.
Call Υm,β : ξm,β → (2SS2 , ωst) the symplectic trivialisation associated to this frame.
It is given by Υm,β(Z) =
√
hm,β(η(Z), λ(Z)) ∈ R2.
Proof. To find a basis for ξm,β, we set H˜m,β := H+aHV and X˜
m,β := X+aXV ,
for some aH , aX ∈ R. Imposing λgm,β(H˜m,β) = 0, we get
0 = mλ(H + aHV )− pi∗β(H + aHV ) + τ(H + aHV ) = 0− βx(xv) + aH .
Hence, we have aH = βx(xv). In the same way we find aX = βx(v) −m. In order
to turn this basis into a symplectic one, we compute
ω′m(H˜
m,β, X˜m,β) = ω′m(H + aHV,X + aXV )
= ω′m(H,X) + aXω
′
m(H,V ) + aHω
′
m(V,X)
= −(−f) + aX · (−m) + aH · 0
= hm,β.
Thus
(
Hˇm,β, Xˇm,β
)
, as defined in the statement of this lemma, is a symplectic basis.
We now find the coordinates of Z = a1Hˇm,β + a2Xˇm,β with respect to this basis:
η(Z) = η(a1Hˇm,β + a2Xm,β) = a1η(Hˇm,β) + a2η(Xm,β) =
a1√
hm,β
+ a2 · 0.
In the same way, λ(Z) = a
2√
hm,β
, so that (a1, a2) =
√
hm,β(η(Z), λ(Z)).
When m = 0, the objects defined above reduce to
λg0,β = −pi∗β + τ R0,β = V,
Hˇ0,β =
1√
f
(
H + βx(xv)V
)
X0,β =
1√
f
(
X + βx(v)V
)
.
7.1. CONTACT FORMS ON LOW ENERGY LEVELS 87
In particular, λg0,β is an S
1connection form on SS2 with curvature σ. Therefore, we
can think of m 7→ λgm,β as a deformation of an S1connection with positive curvature
through contact forms. This observation leads to the following result.
Lemma 7.4. There exists a diffeomorphism Fm,β : SS
2 → SS2 and a real func
tion qm,β : SS
2 → R such that
F ∗m,βλ
g
m,β = e
qm,βλg0,β. (7.6)
The family of diffeomorphisms is generated by the vector field
Zm,β :=
Hˇm,β√
hm,β
=
1
hm,β
(
H + βx(xv)V
)
∈ kerλgm,β. (7.7)
The map
[0,mβ) −→ C∞(SS2,R)
m 7−→ qm,β
is smooth and admits the Taylor expansion at m = 0
qm,β =
1
2f
m2 + o(m2). (7.8)
Proof. We apply Gray Stability Theorem (see for example [Gei08, Theorem
2.2.2]) to the family m 7→ λgm,β and get the equation
ıZm,βω
′
m = −
d
dm
λgm,β = −λ, with Zm,β ∈ kerλgm,β, (7.9)
for Zm,β and the equation
∂
∂m
qm,β =
(
d
dm
λgm,β
)
(Rm,β)Fm,β =
m
hm,β ◦ Fm,β
q0,β = 0
(7.10)
for the function qm,β. There exists a unique pair (Z
m,β, qm,β) satisfying such rela
tions. By Lemma 7.3, we know that Zm,β = aHHˇ
m,β + aXXˇ
m,β. Using the fact
that (Hˇm,β, Xˇm,β) is an ω′msymplectic basis, we get
aH = ıZm,βω
′
m(X
m,β) = −λ(Xm,β) = − 1√
hm,β
aX = −ıZm,βω′m(Hˇm,β) = λ(Hˇm,β) = 0.
Since the function (m, z) 7→ ddmqm,β(z) is smooth on [0,mβ) × SS2, the same
is true for (m, z) 7→ qm,β(z). Therefore, the map m 7→ qm,β is smooth in the
C∞topology and we can expand it at m = 0. From (7.10) we see that q0,β = 0,
∂
∂mq0,β = 0 and
∂2
∂m2
q0,β =
1
h0,β◦F0,β =
1
f . Thus, (7.8) follows.
After this preliminary discussion we are ready to prove the results on periodic
orbits we outlined in the introduction. In the next section we start with the expan
sion in the parameter m of the action function Sm : SS
2 → R defined by Ginzburg
in [Gin87]. We point out that this results holds true also for magnetic systems on
surfaces of higher genus.
88 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Proposition 7.5. There exists a smooth family of smooth functions m 7→ Sm,
where Sm : SS
2 → R, such that
• the critical points of Sm are the support of those periodic orbits of Xm which
are close to a vertical fibre;
• the following expansion holds
Sm = 2pi +
pi
f
m2 + o(m2). (7.11)
Corollary 7.6. If x ∈ S2 is a nondegenerate critical point of f : S2 → R,
then there exists a smooth family of curves m 7→ γm, such that
• γ0 winds uniformly once around SxS2 in the positive sense;
• the support of γm is a periodic orbit for Xm.
Remark 7.7. Corollary 7.6 was already known by the experts, even if there is
no explicit proof in the literature. Ginzburg mentions it in its survey paper [Gin96]
in the paragraph after Remark 3.5 on page 136. Moreover, Castilho, using Arnold’s
approach of the guiding centre approximation [Arn86], constructs a local normal
form for a magnetic system on low energy levels which is similar to the one we found
in Lemma 7.4 (see [Cas01, Theorem 3.1]). It could be used to prove a local version
of Proposition 7.5 and, hence, the corollary.
In Section 7.3 and Section 7.4, we are going to give two independent proofs of
the following theorem.
Theorem 7.8. The 1form λgm,β ∈ Ω1(SS2) is dynamically convex for small m.
Combining this theorem with the abstract results contained in Chapter 6 we get
the result below.
Corollary 7.9. Let (S2, g, σ) be a symplectic magnetic system. If m is small
enough, then the magnetic flow on Σm has either two or infinitely many periodic
orbits. In the first case, the two orbits are noncontractible. Therefore, the second
alternative holds if there exists a prime contractible periodic orbit.
If we suppose in addition that Σm is a nondegenerate level, then
a) there is an elliptic periodic orbit (hence the system is not ergodic);
b) there is a Poincare´ section of disctype.
Remark 7.10. The existence of two periodic orbits on low energy levels of sym
plectic magnetic systems is not new. In [Gin87, Assertion 3], Ginzburg proves the
existence of 2 periodic orbits on S2 and of 3 periodic orbits on surfaces of genus at
least 1. Such orbits are close to the fibres of Σm → M . Furthermore, when the
energy level is nondegenerate, he improves these lower bounds by finding at least
4− eM periodic orbits, 2 of which are elliptic (see the characterisation of the return
map as the Hessian of the action at the end of page 104 in [Gin87]).
Remark 7.11. Example 4.13 exhibits energy levels where all orbits are periodic
and no prime periodic orbit is contractible. Theorem 1.3 in [Sch11] (see also Section
7.2. THE EXPANSION OF THE GINZBURG ACTION FUNCTION 89
7.6 below) shows that there exists an energy level with only two periodic orbits close
to the fibres. It is still an open problem to find an energy level on a nonexact
magnetic system on S2 with exactly two periodic orbits. We point out that in the
class of exact magnetic systems such example has been found. Indeed, Katok showed
in [Kat73] that there are Randers metrics on S2 with only two closed geodesics (see
also [Zil83]) and G. Paternain in [Pat99b, Section 2] showed that every geodesic
flow of a Randers metric arises, up to time reparametrisation, on a supercritical
energy level of some exact magnetic field on S2.
Combining Corollary 7.6 with Corollary 7.9, we can formulate a sufficient con
dition for the existence of infinitely many periodic orbits.
Corollary 7.12. Let (S2, g, σ = fµ) be a symplectic magnetic system. If the
function f : S2 → R has three distinct critical points xmin, xMax and xnondeg such
that xmin is an absolute minimiser, xMax is an absolute maximiser and xnondeg is
nondegenerate, then there exist infinitely many periodic orbits of the magnetic flow
on every sufficiently small energy level.
In Section 7.5 we prove Theorem I, establishing a dichotomy between short and
long periodic orbits. We rephrase it here for the convenience of the reader.
Theorem 7.13. Suppose (g, σ) ∈ Mag(S2) and σ is symplectic. Given ε > 0
and a positive integer n, there exists mε,n > 0 such that for every m < mε,n the
projection pi(γ) of a periodic prime solution γ : TT of Xm either is a simple curve
on S2 with length in ( 2pi−εmax fm,
2pi+ε
min fm) or has at least n selfintersections and length
larger than mε .
This result was inspired to us by [HS12], where a completely analogous state
ment for Reeb flows on convex hypersurface in C2 close to the round S3 is proved.
In that paper the authors show that short orbits are unknotted, the linking number
between two short orbits is 1, while the linking number between shorts and long
orbits goes to infinity as the hypersurface approaches the round S3. In view of
Corollary 7.18, it is likely that Theorem 7.13 can also be proven as a corollary of
Theorem 1.6 in [HS12] by making explicit the relation between linking numbers on
S3 and selfintersections of the projected curve on S2.
Finally, the aim of Section 7.6 is to prove Proposition J regarding rotationally
symmetric magnetic flows. In such proposition we formulate a sufficient condition for
having infinitely many periodic orbits and a sufficient condition for having exactly
two short periodic orbits, on every low energy level.
7.2. The expansion of the Ginzburg action function
We recall the definition of Sm. Set λˇ0 := λ
g
0,β, λˇm := e
qm,β λˇ0. Let Rˇ
m = Rλˇm
be the associated Reeb vector field. For each (x, v) ∈ SS2 we construct a local
Poincare´ section as follows. If y is a curve on S2, denote by P yt : Sy(0)S
2 → Sy(t)S2
the parallel transport with respect to the S1connection λˇ0. Let B
2
δ ⊂ S2 be a small
90 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
geodesic ball centered at x. For every x′ ∈ B2δ , let yxx′ : [0, 1] → S2 be the geodesic
connecting x to x′ and define
i(x,v) : B
2
δ −→ SS2
x′ 7−→
(
x′, P
yx
x′
1 (v)
)
.
The map i(x,v) is a local section of the S
1bundle SS2 such that i(x,v)(x) = (x, v).
Therefore, it is transverse to Rˇm for every m sufficiently small and there exists a
first positive return time tm(x, v) such that Φ
Rˇm
tm(x,v)
= i(x,v)(x
′
m(x, v)), for some
x′m(x, v) ∈ B2δ . Define the path γˇm(x, v) : [0, 1]→ SS2 by
γˇm(x, v)(r) :=
(
yxx′m(x,v)(r), P
yx
x′m(x,v)
r (v)
)
and define γˇm(x, v) to be the inverse path. Notice that
d
dr γˇm(x, v) ∈ ker λˇ0 = ker λˇm.
The loop γm(x, v) : [0, 2]→ SS2 is obtained by concatenation
γm(x, v)(r) =
ΦR
m
tm(x,v)r
(x, v) for r ∈ [0, 1],
γˇm(x, v)(r − 1) for r ∈ [1, 2].
(7.12)
Denote by Cˇ∞(T2, SS2) the space of piecewise smooth loops of period 2 in SS2 and
notice that the family of maps m 7→
(
γm : SS
2 → Cˇ∞(T2, SS2)
)
is smooth. Finally,
define Sm as the composition of the action functional associated with λˇm and the
above embedding of SS2 inside the space of loops:
Sm := Aλˇm ◦ γm : SS2 −→ R
(x, v) 7−→
∫
T2
γm(x, v)
∗λˇm = tm(x, v) +
∫
T1
γˇm(x, v)
∗λˇm
= tm(x, v).
It is proven in [Gin96, Lemma 3, page 104] that, if (x, v) is a critical point of Sm,
then ΦRˇ
m
tm(x,v)
= (x, v) and, therefore, (x, v) is in the support of a periodic orbit.
Notice that since the family m 7→ Sm is smooth, it admits an approximating
Taylor expansion truncated at any order. To find such expansion, we observe that
λˇm =
λˇ0
pi∗ρm +o(m
2) by Lemma 7.4, where ρm : S
2 → R is given by ρm(x) := 1− m22f(x) .
Thus, we prove Proposition 7.5 in two steps:
i) we find an expansion for S′m : SS2 → R, the action function for the 1form
λˇ′m :=
λˇ0
pi∗ρm and we see that is equal to the desired expansion of Sm;
ii) we show that Sm − S′m = o(m2).
For the first step we need the lemma below.
Lemma 7.14. Let pi : E → M be an S1bundle over a closed orientable surface.
Let τ be an S1connection on E with positive curvature form σ. Fix ρ : M → (0,+∞)
a positive function and define the contact form τρ :=
τ
pi∗ρ ∈ Ω1(E). Then, the Reeb
vector field of τρ splits as
R
τρ
(x,v) = −LH(x,v)
(
(Xρ)x
)
+ ρ(x)V(x,v), (7.13)
7.2. THE EXPANSION OF THE GINZBURG ACTION FUNCTION 91
where LH is the horizontal lift with respect to ker τ and Xρ ∈ Γ(M) is the σ
Hamiltonian vector field associated to ρ (namely, ıXρσ = −dρ).
Proof. Since σ is positive, τ is a contact form. Hence, also τρ, which is obtained
multiplying τ by a positive function, is a contact form as well. Without loss of
generality we write R
τρ
(x,v) = L
H
(x,v)(Z(x,v)) + a(x, v)V(x,v), where Z(x, v) ∈ TxM and
a(x, v) ∈ R are to be determined. Imposing that 1 = τρ(Rτρ), we obtain that
a(x, v) = ρ(x). Imposing that 0 = ıRτρdτρ, we get
0 = ıLH(Z)+pi∗ρV
(−pi∗σ
pi∗ρ
− pi∗
(
dρ
ρ2
)
∧ τ
)
= pi∗
(−ıZσ + dρ
ρ
)
− pi∗
(
dρ(Z)
ρ2
)
τ.
This implies that −ıZσ + dρ = 0 and dρ(Z)ρ2 = 0. The first condition implies that
−Z is the σHamiltonian vector field with Hamiltonian ρ. Since the Hamiltonian
function is a constant of motion, the second condition is also satisfied and the lemma
is proven.
Let us proceed to the proof of the proposition following steps i) and ii).
Proof of Proposition 7.5. Let us find the expansion for m→ S′m. Since we
already know that this expansion will be uniform, we can fix the point (x, v) ∈ SS2
and expand the function m 7→ S′m(x, v). We denote with a prime all the objects
associated with λˇ′m. Denote by (x˜(t), v˜(t)) the flow line of R′m going through (x, v)
at time 0. Consider P x˜t′m(x,v)
(v) ∈ Sx′m(x,v)S2, the parallel transport of v along x˜. The
angle between P x˜t′m(x,v)
(v) and γˇ′m(x, v)(1) is equal to the integral of the curvature
σ on a disc bounding the concatenated loop x˜ ∗ pi (γˇ′m(x, v)). Since this loop is
contained inside a ball of radius d(x, x′m(x, v)), we can bound the area of the disc by
some constant times d(x, x′m(x, v))2. However,
d
dt x˜ = Xρm = −m2X2/f implies that
d(x, x′m(x, v)) ≤ tm(x, v) ·m2
∥∥X2/f∥∥. Hence, the area of the disc goes to zero faster
than m3. As a consequence, the angle between P x˜t′m(x,v)
(v) and γˇ′m(x, v)(1) is of
order o(m3). Consider the continuous loop γ˚′m(x, v) obtained by the concatenation
of three paths:
• the path (x˜(t), P x˜t (v));
• a path contained in the fibre over x′m(x, v) and connecting P x˜t′m(x,v)(v) with
γˇ′m(x, v)(1) following the shortest angle;
• the path γˇ′m(x, v).
Since pi(˚γ′m(x, v)) = pi(γ′m(x, v)), we can consider the increment in the angle between
γ˚′m(x, v) and γ′m(x, v) along this loop. As γ˚′m(x, v)(0) = γ′m(x, v)(0), this angle will
be a multiple of 2pi. For m = 0 we readily see that the angle is 2pi, hence it remains
2pi by continuity. However, the derivative of the increment of this angle is
• t′m(x, v)ρm(x), over x˜;
• of order o(m3), over the path contained in the fibre of x′m(x, v);
• zero, over pi(γˇ′m(x, v)).
92 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Putting things together, we find 2pi = t′m(x, v)ρm(x) + o(m3), which implies Step i):
S′m(x, v) = t
′
m(x, v) = 2pi
(
1− m
2
2f(x)
)
+ o(m3).
For Step ii) we observe that there exists a smooth path m 7→ λˇ′′m ∈ Ω1(SS2)
such that λˇm = λˇ
′
m + m
3λˇ′′m. As a consequence, Rλˇm = Rλˇ
′
m + m3R′′m, for some
smooth family m 7→ R′′m ∈ Γ(SS2). This implies that tm(x, v) − t′m(x, v) = o(m2)
and, hence, Sm = tm = t
′
m + o(m
2) = S′m + o(m2). This establish Step ii) and the
entire proposition.
We now have all the tools to prove the corollary.
Proof of Corollary 7.6. Applying the proposition we just proved, we find
that Sm = 2pi + m
2Sˇm, where Sˇ0 =
pi
f . Therefore, the critical points of Sm are
the same as the critical points of Sˇm. Notice that the critical points of Sˇ0 are the
vertical fibres SxS
2, where x ∈ S2 is a critical point of f . Let x be a nondegenerate
critical point of f and consider a tubular neighbourhood T2pi × Bδ of SxS2 with
coordinates (ψ, z). For every ψ ∈ T2pi consider the restriction of Sˇm to {ψ} × Bδ
and call it Sˇψm. Since x is a nondegenerate critical point, the functions Sˇ
ψ
0 have
0 ∈ Bδ as nondegenerate critical point. Therefore, by the inverse function theorem,
for every sufficiently small m there is a path zm : T2pi → Bδ such that zm(ψ) is
the unique critical point of Sψm. We claim that γm(ψ) := (ψ, zm(ψ)) ∈ T2pi × Bδ is
a critical point for Sˇm. Indeed, the function Sm ◦ γm : T2pi → R has at least one
critical point ψ∗, so that dγm(ψ∗)Sm
(
d
dψγm(ψ∗)
)
= 0. Since ddψγm(ψ∗) is transverse
to {0} × Tzm(ψ∗)Bδ ⊂ Tγm(ψ∗)T2pi ×Bδ and
dγm(ψ∗)Sm
∣∣
{0}×Tzm(ψ∗)Bδ
= dzm(ψ∗)S
ψ∗
m = 0,
we have that γm(ψ∗) is a critical point for Sm. However, the critical points for Sm
comes in S1families since they correspond to periodic orbits of Rˇm. This implies
the claim and finishes the proof of the corollary.
Remark 7.15. The approximation of the magnetic flow with the Reeb flow of
Rλˇ
′
m is called the guiding centre approximation. A precise formulation can be found
in [Arn97]. It was used by Castilho in [Cas01] to prove a theorem on region
of stabilities for the magnetic flow via Moser’s invariant curve theorem [Mos77].
Recently, Raymond and Vu˜ Ngo.c [RVN13] employed this approach to study the
semiclassical limit of a magnetic flow with low energy close to a nondegenerate
minimum of f .
7.3. Contactomorphism with a convex hypersurface
The main goal of this section is to construct a convex hypersurface in C2 which
is a contact double cover of (SS2, λgm,β). Example 6.4 will then imply that λ
g
m,β is
dynamically convex and give a first proof of Theorem 7.8.
Proposition 7.16. If m ∈ [0,mβ), there exists a double cover pm,β : S3 → SS2
and an embedding υm,β : S
3 → C2 bounding a region starshaped around the origin
7.3. CONTACTOMORPHISM WITH A CONVEX HYPERSURFACE 93
and such that p∗m,βλ
g
m,β = −υ∗m,βλst. Furthermore, as m goes to zero, υm,β tends
in the C2topology to the embedding of S3 as the Euclidean sphere of radius 2. In
particular, υm,β is a convex embedding for m sufficiently small.
We construct the double cover pm,β in three steps. Let (S
2, g0, µ0) be the mag
netic system on the round sphere of radius 1 given by the area form. We denote by
SS20 the unit sphere bundle, by 0 the rotation by pi/2 and by τ0 the vertical form
associated with the metric g0. We have already seen in Lemma 7.4 that there exists
a contactomorphism Fm,β between e
qm,βλg0,β and λ
g
m,β. Our next task is to relate
λg0,β with τ0. For this purpose, we need the following proposition due to Weinstein
[Wei75]. For a proof we refer to [Gui76, Appendix B].
Proposition 7.17. Suppose Ei → S2, with i = 0, 1, are two S1bundles endowed
with S1connection forms τi ∈ Ω1(Ei). Call σi ∈ Ω2(S2) their curvature forms and
suppose they are both symplectic and such that∣∣∣∣∫
S2
σ0
∣∣∣∣ = ∣∣∣∣∫
S2
σ1
∣∣∣∣ , (7.14)
Then, there is an S1equivariant diffeomorphism B : E0 → E1 such that B∗τ1 = τ0.
Thanks to the normalisation [σ] = 4pi and the fact that λg0,β is an S
1connection
form on SS2, we get the following corollary.
Corollary 7.18. There is an S1equivariant diffeomorphism Bβ : SS
2
0 → SS2
such that B∗βλ
g
0,β = τ0.
What we have found so far tells us that we only need to study the pullback of τ0
to S3. This will be our next task. The ideas we use are taken from [CO04, HP08].
We identify C2 with the space of quaternions by setting 1 := (1, 0), i := (i, 0),
j := (0, 1) and k := (0, i). With this choice left multiplication by i corresponds to the
action of Jst. Let υ : S
3 → C2 be the inclusion of the unit Euclidean sphere. Identify
the Euclidean space R3 with the vector space spanned by i, j,k endowed with the
restricted inner product. We think the round sphere (S2, g0) as embedded in this
version of the Euclidean space. Thus, the unit sphere bundle SS20 is embedded in
R3 × R3 as the pair of vectors (u1, u2) such that u1, u2 ∈ S2 and gst(u1, u2) = 0.
If z = (u1, u2) ∈ SS20 ⊂ R3 × R3 and Z = (v1, v2) ∈ TzSS20 ⊂ R3 × R3, then
(τ0)z(Z) = g0
(
v2 − gst
(
v2, u1
)
u1, 0u1(u2)
)
= gst
(
v2, 0u1(u2)
)
(7.15)
as a consequence of the relation between the Levi Civita connections on S2 and R3.
For any U ∈ S3, we define a map CU : R3 → R3 using quaternionic multiplication
and inverse by CU (U
′) = U−1U ′U . The quaternionic commutation relations and the
compatibility between the metric and the multiplication tell us that CU restricts to
an isometry of S2. Hence, dCU yields a diffeomorphism of the unit sphere bundle
onto itself given by (u1, u2) 7→ du1CU (u2) = (CU (u1), CU (u2)). Moreover, since CU
is an isometry, (dCU )
∗τ0 = τ0.
We are now ready to define the covering map p0 : S
3 → SS20 . It is given by
p0(U) := diCU (j). Let us compute the pullback of τ0 by p0.
94 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Proposition 7.19. The covering map p0 relates τ0 and λst in the following way:
p∗0τ0 = −4υ∗λst. (7.16)
Proof. First of all we prove that both sides of (7.16) are invariant under right
multiplication. For every U ∈ S3, we define RU : S3 → S3 as RU (U ′) := U ′U . Thus,
the identity p0 ◦RU = dCU ◦ p0 holds. Let us show that p∗0τ0 is right invariant:
R∗U
(
p∗0τ0
)
= (p0 ◦RU )∗τ0 = (dCU ◦ p0)∗τ0 = p∗0
(
(dCU )
∗(τ0)
)
= p∗0τ0.
On the other hand, υ∗λst is also right invariant:(
R∗U (υ
∗λst)
)
U ′(W ) =
(
υ∗λst
)
RU (U ′)
(dRUW ) =
1
2
gst
(
(iU ′)U,WU
)
=
1
2
gst
(
iU ′,W
)
=
(
υ∗λst
)
U ′(W ),
where we used that RU : C2 → C2 is an isometry.
Therefore, it is enough to check equality (7.16) only at the point 1. A generic
element W of T1S
3 can be written as si + wj = si + jw, where w := w11 + w2i and
w := w11− w2i with s, w1 and w2 real numbers. On the one hand,
(υ∗λst)1(W ) =
1
2
gst
(
i1,W
)
=
1
2
gst
(
i, si + wj
)
=
s
2
. (7.17)
On the other hand, we have that d1p0(W ) = (iW − W i, jW − W j). From the
definition of τ0 we see that we are only interested in the second component:
jW −W j = j(si + jw)− (si + wj)j = −2sk + (w − w) = −2sk + 2w2i.
Now we apply formula (7.15) with (u1, u2) = (i, j) and v2 = −2sk + 2w2i. In this
case 0u1 is left multiplication by i, so that 0u1(u2) = ij = k and we find that
gst
(− 2sk + 2w2i,k) = −2s. (7.18)
Comparing (7.17) with (7.18) we finally get (p∗0τ0)1 = −4(υ∗λst)1.
Putting things together, we arrive at the following intermediate step.
Proposition 7.20. There exists a covering map pm,β : S
3 → SS2 and a real
function q̂m,β : S
3 → R such that
p∗m,βλ
g
m,β = −4eq̂m,βυ∗λst.
Moreover the function q̂m,β tends to 0 in the C
∞topology as m goes to zero.
Proof. Lemma 7.4 gives us Fm,β : SS
2 → SS2 and qm,β : SS2 → R. 7.18 gives
us Bβ : SS
2
0 → SS2. If we set pm,β := Fm,β ◦Bβ ◦ p0 : S3 → SS2, then
p∗m,βλ
g
m,β = p
∗
0
(
B∗β(F
∗
m,βλ
g
m,β)
)
= p∗0
(
B∗β(e
qm,βλg0,β)
)
= p∗0(e
qm,β◦Bβτ0)
= −4eqm,β◦Bβ◦p0υ∗λst.
7.4. A DIRECT ESTIMATE OF THE INDEX 95
Defining q̂m,β := qm,β ◦Bβ ◦ p0 we only need to show that q̂m,β goes to 0 in the C∞
topology. This is true since, by Lemma 7.4, the same holds for qm,β.
Proof of Proposition 7.16. The final step in the proof is to notice that con
tact forms of the type ρυ∗λst ∈ Ω1(S3) with ρ : S3 → (0,+∞) arise from em
beddings of S3 in C2 as the boundary of a starshaped domain. To see this, define
υ√ρ : S3 ↪→ C2 as υ√ρ(z) :=
√
ρ(z)υ(z). A computation shows that υ∗√ρλst = ρυ
∗λst.
Using this observation we see that p̂∗m,βλ
g
m,β = −υ∗m,βλst with υm,β := υ√ρm,β
and ρm,β := 4e
q̂m,β . For small m, ρm,β is C
2close to the constant 4 and, therefore,
the embedding υm,β is C
2close to the sphere of radius 2. This shows that υm,β(S
3)
for small m and the proof is complete.
Remark 7.21. In general the convexity of the embedding υρ can be verified
defining the function Qρ : C2 → [0,+∞), where Qρ(z) := z2/ρ( zz). Then,
υ√ρ(S3) = {Qρ = 1} and υ√ρ(S3) is convex if and only if the Hessian of Qρ is
positive.
7.4. A direct estimate of the index
In this subsection we present an alternative proof of Theorem 7.8 showing that
λgm,β is dynamically convex via a direct estimate of the index, as prescribed by
Definition 6.1. The advantage of this method is that it generalises to systems on
surfaces of genus at least two as we discuss in Subsection 7.4.1. We can think of this
proof as the magnetic analogue of what Harris and Paternain did for the geodesic
flow, where the pinching condition on the curvature plays the same role as assuming
m small and σ symplectic. After writing this proof, we discovered a similar argument
in [HS12, Section 3.2].
To compute the index, we consider for each z = (x, v) ∈ SS2 the path
Ψm,βz (t) := Υ
m,β
Φm,βt (z)
◦ dzΦm,βt ◦
(
Υm,βz
)−1 ∈ Sp(1).
We define the auxiliary path Bm,βz (t) := Ψ˙
m,β
z (Ψ
m,β
z )−1 ∈ gl(2,R). The bracket
relations (2.7) for (X,V,H) allow us to give the following estimate for this path.
Lemma 7.22. We can write Bm,βz = Jst + ρ
m,β
z , where ρ
m,β
z : R → gl(2,R) is a
path of matrices, whose supremum norm is of order O(m) uniformly in z as m goes
to zero. In other words, there exist m > 0 and C > 0 not depending on z, but only
on sup f , inf f , ‖df‖ and ‖β‖, such that
∀m ≤ m, ‖ρm,βz ‖ ≤ Cm. (7.19)
Proof. For any point z ∈ SS2 and for any vector −→a0 = (a10, a20) ∈ R2 we have a
path −→a = (a1, a2) : R→ R2 defined by the relation
−→a = Ψm,βz −→a 0. (7.20)
Using the definition of Ψm,βz , we see that
−→a satisfies
Z
−→a0
z (t) := dzΦ
m,β
t
(
a10Hˇ
m,β
z + a
2
0Xˇ
m,β
z
)
= a1(t)Hˇm,β
Φm,βt (z)
+ a2(t)Xm,β
Φm,βt (z)
.
96 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
If we differentiate with respect to t the identity
a10Hˇ
m,β
z + a
2
0Xˇ
m,β
z = dΦm,βt (z)
Φm,β−t Z
−→a0
z (t), (7.21)
we get the following differential equation for −→a :
0 = a˙1(t)Hˇm,β
Φm,βt (z)
+ a˙2(t)Xˇm,β
Φm,βt (z)
+ a1(t)
[
Rm,β, Hˇm,β
]
Φm,βt (z)
+ a2(t)
[
Rm,β, Xˇm,β
]
Φm,βt (z)
.
(7.22)
To estimate the first Lie bracket, we observe that m 7→ [Rm,β, Hˇm,β] is a Γ(SS2)
valued map which is continuous in the C0topology since the maps m 7→ Rm,β
and m 7→ Hˇm,β are continuous in the C1topology. As a result, we have that[
Rm,β, Hˇm,β
]
=
[
R0,β, Hˇ0,β
]
+ ρ, where ρ is an O(m) in the C0topology, as m goes
to zero. Furthermore,[
R0,β, Hˇ0,β
]
=
[
V,
1√
f
(
H + (β ◦ )V )] = 1√
f
(
[V,H] + V (β ◦ )V
)
=
1√
f
(
−X − βV
)
= −X0,β
= −Xm,β + ρ′.
Putting things together,
[
Rm,β, Hˇm,β
]
= −Xm,β + ρ1, where ρ1 is an O(m) in the
C0topology. In a similar way we find that
[
Rm,β, Xm,β
]
= Hˇm,β + ρ2. Substituting
these expressions for the Lie brackets inside (7.22), we find
a˙1Hˇm,β + a˙2Xˇm,β = a1Xˇm,β − a2Hˇm,β − a1ρ1 − a2ρ2.
Applying the trivialisation Υm,β, we get
−˙→a = (Jst + ρm,βz )−→a , (7.23)
where ρm,βz ∈ gl(2,R) is of order O(m).
On the other hand, differentiating Equation (7.20), we have
−˙→a = Ψ˙m,βz −→a 0 = Ψ˙m,βz
(
Ψm,βz
)−1−→a = Bm,βz −→a . (7.24)
Thus, comparing (7.23) and (7.24), we finally arrive at Bm,βz = Jst + ρ
m,β
z .
The previous lemma together with the following proposition reduces dynamical
convexity to a condition on the period of Reeb orbits. First, we need the following
notation. For each Z ∈ Γ(N) we call T0(Z) the minimal period of a closed con
tractible orbit of ΦZ . We set T0(Z) = 0, if Φ
Z has a rest point. We remark that the
map Z 7→ T0(Z) is lower semicontinuous with respect to the C0topology.
Proposition 7.23. Let C be the constant contained in (7.19). If the inequality
2pi
T0(Rm,β)
< 1− Cm, (7.25)
is satisfied, then λgm,β is dynamically convex.
7.4. A DIRECT ESTIMATE OF THE INDEX 97
Proof. Let γ be a contractible periodic orbit for Rm,β with period T and such
that γ(0) = z. Consider Ψm,βz
∣∣
[0,T ]
∈ SpT (1) defined as before and fix a u ∈ R2 \{0}.
If θΨ
m,β
z
u is defined by Equation (2.29), we can bound its first derivative by means of
Lemma 7.22, as follows:
θ˙Ψ
m,β
z
u =
gst
(
Ψ˙m,βz u, JstΨ
m,β
z u
)
Ψm,βz u2
=
gst
(
Bm,βz Ψ
m,β
z u, JstΨ
m,β
z u
)
Ψm,βz u2
=
gst
(
(Jst + ρ
m,β
z )Ψ
m,β
z u, JstΨ
m,β
z u
)
Ψm,βz u2
=
Ψm,βz u2 + gst
(
ρm,βz Ψ
m,β
z u, JstΨ
m,β
z u
)
Ψm,βz u2
≥ 1− ‖ρm,βz ‖.
Hence, we can estimate the normalised increment in the interval [0, T ] by
∆(Ψm,βz
∣∣
[0,T ]
, u) =
1
2pi
∫ T
0
θ˙Ψ
m,β
z
u (t)dt ≥ (1− Cm)
T
2pi
.
Therefore, by Remark (6.2), µlCZ(γ) ≥ 3 provided (1 − Cm) T2pi > 1. Asking this
condition for every contractible periodic orbit is the same as asking Inequality (7.25)
to hold. The proposition is thus proved.
We are now ready to reprove Theorem 7.8.
Second proof of 7.8. Thanks to Proposition 7.23, it is enough to show that
Inequality (7.25) holds for small m. First, we compute the periods of contractible
orbits for R0,β = V . A loop going around the vertical fibre k times with unit
angular speed has period 2pik and is contractible if and only if k is even. Hence,
T0(R
0,β) = 4pi.
Using the lower semicontinuity of the minimal period, we find that
lim sup
m→0
(
2pi
T0(Rm,β)
+ Cm
)
≤ 2pi
4pi
+ 0 =
1
2
< 1
and, therefore, the inequality is still true for m small enough.
7.4.1. Generalised dynamical convexity. Abreu and Macarini [AM] have
recently defined a generalisation of dynamical convexity to arbitrary contact mani
folds. We give a brief sketch of it in this subsection since it applies to low energy
values of symplectic magnetic systems on surfaces of genus at least two.
Let (N, ξ) be a closed contact manifold and let ν be a free homotopy class of
loops in N . Suppose that the contact homology HCν(N, ξ) of (N, ξ) in the class ν
is well defined and that c1(ξ) is νatoroidal, so that the homology is also Zgraded.
Definition 7.24. Define k− and k+ in Z ∪ {−∞,+∞} by
• k− := inf
k∈Z
{
HCνk (N, ξ) 6= 0
}
, • k+ := sup
k∈Z
{
HCνk (N, ξ) 6= 0
}
. (7.26)
98 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
A νnondegenerate contact form α supporting ξ is called positively νdynamically
convex if k− ∈ Z and µCZ(γ) ≥ k− for every Reeb orbit of α in the class ν. Similarly,
α is called negatively νdynamically convex if k+ ∈ Z and µCZ(γ) ≤ k+ for every
Reeb orbit of α in the class ν.
Notice that a contact form α supporting (S3, ξst), that is dynamically convex
according to Definition 6.1, is indeed positively dynamically convex according to Def
inition 7.24, with k− = 3. The new notion of convexity allows Abreu and Macarini
to prove the existence of an elliptic orbit for Reeb flows on BoothbyWang manifolds.
Definition 7.25. A contact manifold (N, ξ) is called BoothbyWang if it admits
a supporting contact form β whose Reeb flow gives a free S1action.
Theorem 7.26 ([AM]). Let (N, ξ = kerβ) be a BoothbyWang contact manifold
and let ν be the free homotopy class of the simple closed orbits of Rβ. Assume that
one of the following hypotheses holds:
• M/S1 admits a Morse function whose critical points have all even index;
• all the iterates of ν are noncontractible.
If G < S1 is a nontrivial finite subgroup, then the Reeb flow of every Ginvariant
positively (respectively, negatively) νdynamically convex contact form α supporting
ξ up to isotopy has an elliptic closed orbit γ representing ν such that µCZ(γ) = k−
(respectively, µCZ(γ) = k+).
Let M be a surface with genus at least two and let σ be a symplectic magnetic
form such that [σ] = 2pieM . In Subsection 4.2.3 we proved that when m is low the
primitives of ω′m of the type λ
g
m,β are contact forms. Consider the cover p : N → SM
which restricts to the standard eM sheeted cover above every fibre SxM . Let νM be
the class on N corresponding to the lift of a curve in SM winding eM  times around
SxM . Then, (N, kerλ
g
0,β) and the class νM satisfy the hypotheses of the theorem (a
curve going around SxM once is a free element in pi1(SM) as can be seen from the
long exact sequence of homotopy groups for the fibration SM → M). Moreover,
Abreu and Macarini computed HCν∗ (N, ker p∗λ
g
m,β) and proved that k+ = 2eM + 1.
Therefore, in order to prove that p∗λgm,β is negatively νdynamically convex,
and be able to apply the theorem, it is enough to show that, for every periodic
orbit (γ, T ) of Rλ
g
m,β homotopic to a curve winding eM  times around the fibres of
SM →M , the following inequality holds:
µCZ(γ) ≤ 2eM + 1. (7.27)
Now we briefly explain how to adapt the argument used on S2 to get Inequality
(7.27). First of all, Rm,β converges to −V (and not to V ), for m tending to 0.
This difference of sign leads to the estimate ‖Bm,βz − (−Jst)‖ = Cm (compare with
Lemma 7.22). We can use this inequality to obtain the upper bound
∆(Ψm,βz
∣∣
[0,T ]
, u) ≤ (−1 + Cm) T
2pi
, ∀u ∈ ξm,β
∣∣
γ(0)
.
7.4. A DIRECT ESTIMATE OF THE INDEX 99
On the other hand, by the lower semicontinuity of the minimal period in the class
νM , we have T > 2pi(eM  − ε), for an arbitrary ε and m < mε. Putting together
these two inequalities we get
I
(
Ψm,βz
∣∣
[0,T ]
) ⊂ (−∞, eM + Cm(eM  − ε) + ε) ⊂ (−∞, eM + 1)
for m < mε small enough. This inclusion implies (7.27), thanks to (2.32).
7.4.2. A geometric estimate of the minimal period. We end this section
by giving a geometric proof of Inequality (7.25) for small values of m giving an
ad hoc proof of the lower semicontinuity of the minimal period in this case. The
construction we present here will turn useful again in Section 7.5.
We consider a finite collection of closed discs D := {Di  Di ⊂ S2} such that
the open discs D˙ := {D˙i} cover S2. We also fix a collection of vector fields of unit
norm Z = {Zi  Zi ∈ Γ(Di), Zi = 1}. Let δ be the Lebesgue number of the cover
D with respect to the Riemannian distance. Finally, let ϕi : SDi → T2pi be the
angular function associated to Zi. We set
Cβ(D,Z)(m) := sup
i
(
sup
Di
dϕi(X)
hm,β
)
and we observe that m 7→ Cβ(D,Z)(m) is locally bounded around m = 0. We now
write down the condition of lower semicontinuity for the minimal period explicitly.
Let ε > 0 be arbitrary. We claim that there exists mε > 0 such that, for m < mε,
the period T of a contractible periodic orbit γ of Rm,β is bigger than 4pi − ε. The
claim will follow from the next two lemmas.
Lemma 7.27. Suppose that (γ, T ) is a periodic orbit for Rm,β and that pi(γ) is
not contained in any Di. If T∗ is any positive real number and m <
2δ inf hm,β
T∗ , then
T > T∗.
Proof. By assumption pi(γ) is not contained in any ball of radius δ. Thus,
2δ ≤ `(pi(γ)) ≤ m
inf hm,β
T <
2δ
T∗
T,
where the second inequality is obtained using Lemma 7.27. Multiplying both sides
by T∗2δ yields the desired conclusion.
Lemma 7.28. Suppose that (γ, T ) is a periodic orbit for Rm,β and that pi(γ) is
contained in some Di0. Let ϕ˜i0 : [0, T ] → R be a lift of ϕi0 ◦ γ
∣∣
[0,T ]
: [0, T ] → T2pi
and let 2piN := ϕ˜i0(T )− ϕ˜i0(0), with N ∈ Z. For every ε′ > 0, there exists m′ε′ > 0
small enough and independent of (γ, T ) such that N ≥ 1 and
T ≥ 2piN(1− ε′). (7.28)
Proof. Using the fundamental theorem of calculus we get
ϕ˜i0(T )− ϕ˜i0(0) =
∫ T
0
dϕi0
dt
dt =
∫ T
0
dϕi0(R
m,β)dt.
100 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Since dϕi0(R
m,β) = mdϕi0
(
X
hm,β
)
+ fhm,β and
f
h0,β
= 1, we have, for m small enough,
dϕi0(R
m,β) ≥ inf
SS2
f
hm,β
−mCβ(D,Z)(m) > 0. (7.29)
This implies that N ≥ 1 and, moreover, that
2piN ≤
∫ T
0
dϕi0(R
m,β)dt ≤
(
sup
SS2
f
hm,β
+mCβ(D,Z)(m)
)
T.
Exploiting fh0,β = 1 again, we see that there exists m
′
ε′ such that, if m < m
′
ε,
2piN(1− ε′) < 2piN
sup
SS2
f
hm,β
+mCβ(D,Z)(m)
≤ T.
We now prove the claim taking mε := min
{
2δ inf hm,β
4pi−ε ,m
′
ε′
}
, where m′ε′ is ob
tained from Lemma 7.28 with a value of ε′ given by the equation 4pi(1−ε′) = 4pi−ε.
If γ is not contained in a Di, the claim follows from Lemma 7.27 with T∗ = 4pi − ε.
If γ is contained in some Di0 , then N is even since γ is contractible. Therefore,
T > 4pi(1− ε′) = 4pi − ε and the claim is proved also in this case.
7.5. A dichotomy between short and long orbits
In this section we prove Theorem 7.13 (and, hence, Theorem I). The statement
will readily follow from the general Corollary 7.30 after we show that
• periodic orbits, whose projection on the base S2 has a fixed number of
selfintersections, have bounded length (Corollary 7.33);
• the projection on S2 of periodic orbits with period close to 2pi is a simple
curve (Lemma 7.34).
The foundational result for our proof is Proposition 1 of [Ban86].
Proposition 7.29 (Bangert). Let Φ :R ×M → M be a C1flow and p ∈ M a
periodic point of prime period T > 0. Then, for every ε > 0 there exist a neigh
bourhood U of Φ in the weak C1topology on C1(M × R,M) and a neighbourhood
U of p in M such that the following is true: If a flow Φ ∈ U has a periodic point
p ∈ U with prime period T then T > ε−1 or there exists a positive integer k such
that kT − T  < ε and the linear map dpΦT : TpM → TpM has an eigenvalue which
generates the group of kth roots of unity.
We can apply this proposition to T periodic flows. A flow is T periodic if and
only if through every point there is a prime periodic orbit whose period is T .
Corollary 7.30. Let Φ be a T periodic flow on a compact manifold M . Then,
for every ε > 0 there exists a C1neighbourhood U of Φ, such that if γ is a periodic
orbit of Φ ∈ U with prime period T , either T > ε−1 or T − T  < ε.
Let us go back to symplectic magnetic systems on low energy levels on S2. We
observed before that for m = 0 the Reeb vector field is equal to the vector field V .
Hence its flow is 2piperiodic and we can apply Bangert result for m small.
7.5. A DICHOTOMY BETWEEN SHORT AND LONG ORBITS 101
Corollary 7.31. For every ε > 0, there exists mε > 0, such that, if m < mε,
every prime periodic orbit γ of Rm,β on SS2 either has period bigger than ε−1 or in
the interval (2pi − ε, 2pi + ε).
In the next lemma we prove that knowing the number of self intersections of
pi(γ) gives a uniform bound on `(pi(γ)). Thanks to Lemma 7.2, this will imply a
bound on the period, as well.
Lemma 7.32. Let x : I → S2 be a closed curve which is a smooth immersion
except possibly at n0 points and has no more than n1 selfintersections. Suppose
moreover that, when defined, the geodesic curvature of x is bounded away from zero.
Namely, min kx > 0. Then,
`(x) ≤ pi(4n1 + 2 + n0) + (n1 + 1)minK volg(S
2)
min kx
(7.30)
Proof. We argue by induction on n1. Suppose n1 = 0, then x bounds a disc D
and we can apply the GaussBonnet formula:∫
x
kx(t)dt = 2pi −
∫
D
KdA−
n0∑
i=1
ϑi.
Since ϑi ≥ −pi, the righthand side is smaller than 2pi+ minK volg(S2)+n0pi. The
lefthand side is bigger than `(x) min kx and dividing both sides by min kx we get
(7.30) in this case.
Take now a curve x with n1 ≥ 1 selfintersections. Then, there exists a smooth
loop x1 inside x. Up to a change of basepoint, x is the concatenation of x1 and x2,
where x2 has at most n1 − 1 selfintersections and n0 + 1 corners. By induction we
can apply inequality (7.30) to the two pieces:
`(x1) ≤ 3pi + minK volg(S
2)
min kx1
,
`(x2) ≤ pi(4n1 − 2 + n0 + 1) + n1minK volg(S
2)
min kx2
.
Since min kx = min{min kx1 ,min kx2} we can substitute it in the denominators above
without affecting the inequality. Adding up the resulting relations we get
`(x) = `(x1) + `(x2) ≤ pi(3 + 4n1 − 2 + n0 + 1) + (n1 + 1)minK volg(S
2)
min kx
,
which is the desired result.
We get immediately the following corollary.
Corollary 7.33. Let γ be a periodic orbit of Rm,β such that pi(γ) has at most
n selfintersections. Then,
`(pi(γ))
m
≤ 4n+ 2 + (n+ 1)minK volg(S
2)
min f
and, therefore,
T ≤ maxhm,β 4n+ 2 + (n+ 1)minK volg(S
2)
min f
=: Cn. (7.31)
102 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Proof. To get the first inequality, we apply Lemma 7.32 to pi(γ), which has
at most n selfintersections and no corners, and observe that kpi(γ) =
f(pi(γ))
m . The
second inequality follows from Lemma 7.2.
If we fix n ∈ N, Corollaries 7.33 and 7.31 show that, for m small enough, a
solution with n selfintersections can exist if and only if its period is close to 2pi.
The next step will be to prove that this happens only if pi(γ) is a simple curve.
Lemma 7.34. For every ε′ > 0, there exists mε′ > 0 such that, if m ≤ mε′ and
γ is a prime periodic orbit whose projection pi(γ) has at least 1 selfintersection, the
period of γ is bigger than 4pi − ε′.
Proof. We use the same strategy of Section 7.4.2 and we consider a collection
of discs and unit vector field (D,Z). The lemma is proven once we show that, if
pi(γ) ⊂ Di, for some Di, then N ≥ 2. We achieve this goal by making a particular
choice of (D,Z). Namely, we take Di contained in some orthogonal chart (x
1
i , x
2
i )
and we take Zi := ∂x1i
/∂x1i .
Suppose without loss of generality that the selfintersection is happening at t = 0.
Thus, we know that there exists t∗ ∈ (0, T ) such that pi(γ(0)) = pi(γ(t∗)). By
precomposing the coordinate chart with an orthogonal linear map we can assume
that the tangent vector at zero is parallel to ∂x1i
, namely that ϕi(0) = 0. We consider
the function x2i ◦ γ : TT → R and compute its derivative. Since the parametrisation
is orthogonal we know that
d
dt
pi(γ) =
∣∣∣∣ ddtpi(γ)
∣∣∣∣
(
cosϕi
∂x1i
∂x1i 
+ sinϕi
∂x2i
∂x2i 
)
.
However, in coordinates we always have
d
dt
pi(γ) =
d
dt
(x1i ◦ γ)∂x1i +
d
dt
(x2i ◦ γ)∂x2i .
Comparing the two expressions we find
d
dt
(x2i ◦ γ) =
∣∣ d
dtpi(γ)
∣∣ sinϕi
∂x2i 
. (7.32)
A point t0 is critical for x
2
i ◦ γ if and only if ϕi(t0) = 0 or ϕi(t0) = pi. Computing
the second derivative at a critical point we get
d2
dt2
∣∣∣
t=t0
(x2i ◦ γ) =
∣∣ d
dtpi(γ)
∣∣
∂x2i 
cosϕiϕ˙i
∣∣∣
t=t0
. (7.33)
Since ϕ˙i > 0 by (7.29), we know that t0 is a strict local minimum if ϕi(t0) = 0 and
a strict local maximum if ϕi(t0) = pi. From the previous discussion we know that
t = 0 is a strict minimum and that x2i ◦ γ(0) = x2i ◦ γ(t∗). Hence, there must exist
two strict maxima t1 ∈ (0, t∗) and t2 ∈ (t∗, T ). At these points ϕi = pi and, thus,
N ≥ 2.
We are now ready to prove Theorem 7.13.
7.6. A TWIST THEOREM FOR SURFACES OF REVOLUTION 103
Proof of Theorem 7.13. Given ε and n, take any ε′ε,n such that the following
inequalities hold for small m:
• ε′ε,n < min{1, ε}, • Cn−1 <
1
ε′ε,n
, • ε′ε,n maxhm,β < ε,
where Cn−1 is the constant defined in (7.31). Define m˜ε,n = min{mε′ε,n ,mε′ε,n},
where mε′ε,n and mε′ε,n are given by Corollary 7.31 and Lemma 7.34, respectively.
If m < m˜ε,n and γ is a periodic orbit with period smaller than
1
ε′ε,n
, using
Corollary 7.31, we find that T ∈ (2pi − ε′ε,n, 2pi + ε′ε,n). Thus T < 4pi − ε′ε,n and, by
Lemma 7.34, pi(γ) is a simple curve in S2. We use Lemma 7.2 to estimate the length
of pi(γ) in this case:
2pi − ε′ε,n
maxhm,β
m ≤ `(pi(γ)) ≤ 2pi + ε
′
ε,n
minhm,β
m. (7.34)
Shrinking the interval [0, m˜ε,n] if necessary, we find some mε,n ≤ m˜ε,n, such that, if
m ∈ [0,mε,n], inequalities (7.34) imply
2pi − ε
max f
m ≤ `(pi(γ)) ≤ 2pi + ε
min f
m,
as required. If the period T is bigger than 1ε′ε,n
> 1ε , then pi(γ) has at least n self
intersections by the condition Cn−1 < 1ε′ε,n and Corollary 7.33. In this case we also
have
`(pi(γ)) ≥ m
ε′ε,n maxhm,β
≥ m
ε
.
This finishes the proof of the theorem.
7.6. A twist theorem for surfaces of revolution
As observed in Remark 7.11, we do not have any example of a contacttype
energy level for a nonexact magnetic system on S2 with exactly two periodic orbits.
This is due to the fact that the only case where we can compute all the trajectories
of the magnetic flow is Example 4.13, where all the orbits are shown to be periodic.
Observe, indeed, that knowing that the flow of Xm ∈ Γ(SS2) is a perturbation
of a flow with exactly two periodic orbits, whose iterations are assumed to be non
degenerate, is not enough to deduce that the magnetic flow has only two periodic
orbits. The only thing that we can deduce is that Xm has exactly two short periodic
orbits [Sch11]. However, long periodic orbits may appear in the perturbed system.
On the other hand, if the perturbation satisfies a suitable twist condition, we
can prove the existence of infinitely many long periodic orbits for the new flow.
In this section we are going to use such perturbative approach on surfaces of revo
lution. Unlike Chapter 5, we consider more general systems of the kind (S2γ , gγ , fµγ),
where f : S2γ → (0,+∞) is any positive function invariant under the rotations around
the axis of symmetry. This means that f depends on the tvariable only and we can
write f : [0, `γ ]→ (0,+∞).
We know that for m small, λgm,βγ is a dynamically convex contact form and,
hence, at least in the non degenerate case, Φm,β
γ
has a Poincare´ section of disctype
104 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
im,βγ : D → SS2γ . Moreover, since Φ0,β
γ
is 2piperiodic, we know that the return
map Fm,βγ : D˚ → D˚ for m = 0 is the identity. Then, if we could find explicitly a
smooth isotopy m 7→ im,βγ of Poincare´ sections, we could try to expand the maps
Fm,βγ in the parameter m around zero to get some information on the dynamics.
However, it is in general a difficult task to construct by hand a Poincare´ section
and the rotational symmetry does not seem to help finding a disc in this case. On
the other hand, we claim that such symmetry allows to find a Poincare´ section of
annulus type, as we will show in the next subsection.
Historically, annuli were the first kind of Poincare´ sections to be studied. The
discovery of such section in the restricted 3body problem [Poi87] led Poincare´ to
formulate his Last Geometric Theorem [Poi12]. It asserts the existence of infinitely
many periodic points of the return map F , if such map satisfies the socalled twist
condition: namely, F extends continuously to the boundary of the annulus and it
rotates the inner and outer circle by different angles. One year later Birkhoff proved
Poincare´’s Theorem [Bir13] and in subsequent research found a section of annulus
type for the geodesic flow of a convex twosphere [Bir66, Chapter VI.10].
In the last subsection we are going to give a condition on γ and f that ensures
that the Poincare´ maps for small values of m are twist. By contrast, the complemen
tary condition will single out a class of magnetic systems whose long periodic orbits
have period or order O(m−2) or higher. It would be interesting to compare this
estimate with the abstract divergence rate coming from Bangert’s proof of Proposi
tion 7.29. A direction of future research would be to look for systems with exactly
2 periodic orbits within this class.
From now on, we consider a fixed profile function γ and the rotationally invariant
primitive βγ = βγθ dθ, so that we can safely suppress the symbols γ and β from the
subscripts and superscripts.
7.6.1. Definition of the Poincare´ map. Take the loop c : [−`, `]/∼ → SS2,
where ∼ is the relation that identifies the boundary of the interval. It is defined in
the coordinates (t, ϕ, θ), as
c(u) :=
(−u,−pi/2, 0), if u < 0,(u, pi/2, pi), if u > 0.
We extend it smoothly for u ∈ {−`, 0, `} and we observe that t(c(u)) = u. Using
the T2piaction given by the rotational symmetry we can move transversally c in SS2
and form an embedded 2torus C : [−`, `]/∼ × T2pi → SS2:
C(u, ψ) :=
(−u,−pi/2, ψ), if u < 0,(u, pi/2, ψ + pi), if u > 0.
In particular, notice that ψ = θ, for u < 0 and ψ = θ + pi, for u > 0. Since m
is sufficiently small, Proposition 5.9 implies that Xmγ := mX + fV has only two
periodic orbits ζ−m and ζ+m supported on latitudes t−m and t+m.
7.6. A TWIST THEOREM FOR SURFACES OF REVOLUTION 105
The sign tells us whether the projection of ζ±m rotates in the same direction as
∂θ or not. The numbers t
±
m satisfy the equation
±mγ˙(t) = f(t)γ(t).
With our choices we have t+m < t
−
m. These two orbits lie inside the image of the torus
C. If we define u−m := −t−m and u+m := t+m, then ψ 7→ C(u±m, ψ) is a reparametrisation
of ζ±m. Hence, the 2torus is divided into the union of two closed cylinders C−m and C+m
with the two orbits as common boundary. Each of the cylinders is a Poincare´ section
for Xm and we would like to compute its first return map F±m : C˚±m → C˚±m. If we look
at the first return map Fm : [−`, `]/∼\{u−m, u+m}×T2pi → [−`, `]/∼×T2pi, we see that
this map swaps the cylinders and F±m = F 2m
∣∣
C±m
. We have F0(u, ψ) = (−u, ψ + pi)
and, as we expected, F±0 (u, ψ) = (u, ψ). We know that Fm is a smooth family of
smooth maps and we claim that it extends on the whole torus to a smooth family of
continuous maps.
Proposition 7.35. The family m 7→ Fm admits an extension to a smooth family
of continuous maps m 7→ F̂m : [−`, `]/∼ × T2pi → [−`, `]/∼ × T2pi.
For the proof of the proposition, we need first to compute the projection of the
differential of the Reeb vector field Rm at ζ±m on the contact distribution ξm.
Lemma 7.36. At a point (t±m,±pi2 ,m θγ(t±m)) we have the identity(
[Hˇm, Rm]
[Xˇm, Rm]
)
=
(
0 fhm −m
H(hm)
h2m
−1 0
)(
Hˇm
Xˇm
)
. (7.35)
Proof. Thanks to Lemma 7.3, the matrix above is given by
√
hm
η([Hˇm, Rm]) λ([Hˇm, Rm])
η
(
[Xˇm, Rm]
)
λ
(
[Xˇm, Rm]
)
. (7.36)
Thus, we have to compute the Lie brackets only up to multiples of V . Below we use
the symbol
.
= between two vectors that are equal up to a multiple of V . Observe
that on the support of ζ±m we have the identities
• β = ±βθ
γ
, • β ◦ = 0, • V (β) = β ◦ = 0, • H = −± ∂˜t, • X = ± ∂˜θ
γ
.
We compute
[Hˇm, Rm] =
[
Hˇm,
m
hm
X +
f
hm
V
]
.
= mHˇm
(
1
hm
)
X +
m
hm
[Hˇm, X] +
f
hm
[Hˇm, V ]
.
= m
1√
hm
H
(
1
hm
)
X +
m
hm
√
hm
[H,X] +
f
hm
√
hm
[H,V ]
.
= m
1√
hm
H
(
1
hm
)
X +
f
hm
√
hm
X.
106 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Similarly,
[Xˇm, Rm] =
[
Xˇm,
m
hm
X +
f
hm
V
]
.
= mXˇm
(
1
hm
)
X +
m
hm
[Xˇm, X] +
f
hm
[Xˇm, V ]
.
= 0 +
m
hm
√
hm
(β −m)[V,X] + f
hm
√
hm
[X,V ]
= −m
2 −mβ + f
hm
√
hm
H
= − 1√
hm
H.
Applying the trivialisation Υm, we get the desired formula.
We now prove a lemma which yields a local version of the proposition, under a
further assumption on the differential of the vector field.
Lemma 7.37. Consider coordinates (z = (x, y), ϕ) on R2 × T2pi and let (r, θ)
be polar coordinates on the first factor. Let Bδ := {r < δ} ⊂ R2 be the open
disc of radius δ and T3δ := Bδ × T2pi be the solid torus with section Bδ. Suppose
that m 7→ Zm ∈ Γ(T3δ) is a smooth family of smooth vector fields and decompose
Zm = Z
R2
m + dϕ(Zm)∂ϕ. Assume that
a) ∀ (m,ϕ), (ZR2m )(0,ϕ) = 0 and the endomorphism dR
2
(0,ϕ)Z
R2
m is antisymmetric;
b) the smooth family of functions am : T2pi → R defined by dR2(0,ϕ)ZR
2
m = amJst is
uniformly bounded from below by a positive constant.
Under these hypotheses, there exists some δ′ < δ and a smooth family of first return
maps on the set
{
(0, y, ϕ)
∣∣ 0 < y < δ′}. Such family extends to a smooth family
of continuous maps Pm :
{
(0, y, ϕ)
∣∣ y < δ′}→ {(0, y, ϕ) ∣∣ y < δ′}.
Proof. We claim that the family of functionsm 7→ dθ(ZR2m ) extends to a smooth
family of continuous functions m 7→ âm on the whole T3δ . Indeed, observe that
d(z,ϕ)θ(·) = ωR2st (z, ·)/r2. As a consequence,
d(z,ϕ)θ(Z
R2
m ) =
ωR
2
st (z, Z
R2
m )
r2
=
ωR
2
st
(
z, dR
2
(0,ϕ)Z
R2
m z + o(r)
)
r2
=
gst
(
z,−JstdR2(0,ϕ)ZR
2
m z
)
r2
+
ωR
2
st
(
z, o(r)
)
r2
= am + o(1).
Since the term o(1) is uniform in m and ϕ, the claim follows.
From the fact that am is bounded away from zero, we deduce that on some small
T3δ′ , d(z,ϕ)θ(Zm) = d(z,ϕ)θ(Z
R2
m ) is also bounded away from zero and, hence, there is a
well defined first return time tm(y, ϕ) for the flow of Zm on {(0, y, ϕ))
∣∣ 0 < y < δ′}.
It is uniquely determined by the equation
pi =
∫ tm(y,ϕ)
0
d
ΦZmt (0,y,ϕ)
θ(ZR
2
m ) dt =
∫ tm(y,ϕ)
0
âm(Φ
Zm
t (0, y, ϕ)) dt.
7.6. A TWIST THEOREM FOR SURFACES OF REVOLUTION 107
Hence, tm(y, ϕ) is defined also for z = 0 andm 7→ tm is a smooth family of continuous
functions because the same is true for m 7→ âm. The required extension of the
Poincare´ map is given by Pm(y, ϕ) = Φ
Zm
tm(y,ϕ)
(0, y, ϕ).
We are now ready to prove the proposition.
Proof of Proposition 7.35. We need only to consider the system close to
the latitudes ζ±m. Since m is small, ζ−m and ζ+m will be near the north and south
pole, respectively. We only analyse the case of the south pole, since for the north
pole we can use a similar argument. Consider the vector field X˚m := bmXˇ
m, where
bm : SS
2 → (0,+∞) is a rotationally invariant function to be determined later.
Notice that such vector field is transverse to C at the south pole, namely when
u = 0. Therefore, the map (xm, um, ψ) 7→ ΦX˚mxm (C(um+u+m, ψ)) yields a welldefined
smooth family of diffeomorphisms between a neighbourhood of the fibre over the
south pole and Bδ × T2pi ⊂ R2 × T2pi. We use these maps to pull back the Reeb
vector field to Bδ × T2pi. We still denote by Rm the pullback.
We show that Rm satisfies the hypotheses of Lemma 7.37. On xm = 0, we have
• ∂xm = X˚m = bmXˇm • ∂um = ∂˜t = −H = −
√
hmHˇ
m,
so that (∂xm , ∂um) is a basis of ξm
∣∣
{xm=0}. Denote by
A :=
(
bm 0
0
√
hm
)
the change of basis matrix. We also have dual relations on ξ∗m:
• dxm =
√
hm
bm
λ • dum = −
√
hm√
hm
η.
Notice that, since bm and
√
hm are rotationally invariant, at {xm = 0}, we have
[∂xm , R
m] = bm[Xˇ
m, Rm] and [∂um , R
m] = −√hm[Hˇm, Rm]. Then, we compute the
matrix of d(0,ψ)(R
m)R
2
in the (xm, um) coordinates
d(0,ψ)(R
m)R
2
=
(
dxm
(
[∂xm , R
m]
)
dxm
(
[∂um , R
m]
)
dum
(
[∂xm , R
m]
)
dum
(
[∂um , R
m]
) )
= A−1
( √
hmλ
(
[Xˇm, Rm]
) √
hmλ
(
[Hˇm, Rm]
)
√
hmη
(
[Xˇm, Rm]
) √
hmη
(
[Hˇm, Rm]
) )A
= A−1
(
0 fhm −m
H(hm)
h2m
−1 0
)
A
=
0 −√hmbm ( fhm −mH(hm)h2m )
bm√
hm
0
Therefore, we need bm√
hm
=
√
hm
bm
(
f
hm
−mH(hm)
h2m
)
. Namely,
bm = +
√
f −mH(hm)
hm
.
108 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
In this case we have
d(0,ψ)(R
m)R
2
=
√
f
hm
−mH(hm)
h2m
Jst
which gives at once that both hypothesis a) and b) are satisfied. Thanks to Lemma
7.37, the Reeb flow admits a smooth family of continuous first return maps on
{(0, um, ψ)  um < δ′} ⊂ C. Since this set is an open neighbourhood of ζ+m in C,
the proposition follows.
Remark 7.38. We do not know if the extended return maps can be taken to be
smooth.
The rotational symmetry of the system implies that Fm can be written in the
form Fm(u, ψ) = (um(u), ψ+ψm(u)). When u and um(u) are not in {−`, 0, `}, then
we can also define the longitude θ of the starting point and the longitude θ+ θm(u)
of the ending point. If u and um(u) have the same sign, then θm(u) = ψm(u), while
if u and um(u) have opposite sign θm(u) = ψm(u) + pi.
The system has also another type of symmetries given by the reflections along
planes in R3 containing the axis of rotation. In coordinates these maps can be
written as (t, θ) 7→ (t, 2θ0−θ), for some θ0 ∈ T2pi. Trajectories of the flow are sent to
trajectories of the flow travelled in the opposite direction. Therefore, if u and um(u)
are not in {−`, 0, `}, then um(um(u)) = u and θm(um(u)) = θm(u). This implies that
ψm(um(u)) = ψm(u) for every u ∈ [−`,+`]/∼. Thus, F 2m(u, ψ) = (u, ψ + 2ψm(u)).
When u is different from 0 and `, and m is suitably small (m will be smaller
and smaller as u is close to 0 or `), u and um(u) have different signs and we can
write an integral formula for ψm(u) = θm(u) + pi. Without loss of generality, we
fix some u < 0 for the forthcoming computation. Let zum : [0, R] → SS2 be the
only solution of the magnetic flow, passing through C(u, 0) at time 0 and through
C(um(u), ψm(u)) at time R. Then, ϕ ◦ zum : [0, R] → [−pi/2, pi/2] is a monotone
increasing function, such that ϕ(zum(0)) = −pi/2 and ϕ(zum(R)) = pi/2. Indeed,
thanks to Equation (5.4), its derivative satisfies the equation
d
dr
ϕ = f − γ˙
γ
m sinϕ
(where we dropped the composition with zum in the notation). Hence, we can
reparametrise zum using the variable ϕ and get the following expression for θm(u):
θm(u) =
∫ pi
2
−pi
2
dθ
dϕ
dϕ =
∫ pi
2
−pi
2
dθ
dr
dr
dϕ
dϕ =
∫ pi
2
−pi
2
m sinϕ
γf − γ˙m sinϕ dϕ, (7.37)
where we substituted dθdr =
m sinϕ
γ , using Equation (5.4) again.
In the next section we expand θm(u) with respect to the variable m around
m = 0. We aim at computing the first nonzero term.
7.6.2. The Taylor expansion of the return map. For small m we have that
t(zum) = u+Ou(m) (the size of Ou(m) depending on u), hence γf is big compared
to γ˙m sinϕ. Thus, we expand the denominator in the integrand in (7.37), up to a
7.6. A TWIST THEOREM FOR SURFACES OF REVOLUTION 109
term of order m2, using the formula for the geometric series:
m sinϕ
γf − γ˙m sinϕ =
m sinϕ
γf
(
1 +
γ˙m sinϕ
γf
)
+ ou(m
2)
= m
1
γf
sinϕ+m2
γ˙
(γf)2
sin2 ϕ+ ou(m
2). (7.38)
Discarding the ou(m
2) remainder, we plug (7.38) into (7.37) and compute the two
resulting integrals separately.
We use integration by parts for the first one∫ pi
2
−pi
2
m
1
γf
sinϕdϕ = m
∫ pi
2
−pi
2
(
d
dt
1
γf
)
dt
dϕ
cosϕdϕ
= m
∫ pi
2
−pi
2
(
d
dt
1
γf
)
m cosϕ
f
cosϕdϕ+ ou(m
2)
= m2
(
d
dt
1
γf
)
1
f
(u) ∫ pi2
−pi
2
cos2 ϕdϕ+ ou(m
2)
=
m2pi
2
(
d
dt
1
γf
)
1
f
(u)+ ou(m2).
For the second integral we readily find∫ pi
2
−pi
2
m2
γ˙
(γf)2
sin2 ϕdϕ = m2
γ˙
(γf)2
(u) ∫ pi2
−pi
2
sin2 ϕdϕ+ ou(m
2)
=
m2pi
2
γ˙
(γf)2
(u)+ ou(m2).
Putting things together we get
θm(u) =
m2pi
2
((
d
dt
1
γf
)
1
f
+
γ˙
(γf)2
)(u)+ ou(m2)
=
m2pi
2
(
−
(
γ˙
γ2f
+
f˙
γf2
)
1
f
+
γ˙
(γf)2
)(u)+ ou(m2)
=
m2pi
2
(
− f˙
γf3
)(u)+ ou(m2).
By Proposition 7.35, we know that ψm(u) = θm(u) + pi is a smooth family of func
tions. Hence, the expansion above translates in an expansion for ψm that holds
on the whole torus and such that the remainder is of order o(m2) uniformly in u.
Indeed, observe that f˙(0) = f˙(`) = 0 and, therefore, the function −f˙/(γf3) extends
smoothly at the poles, taking the value −f¨/f3 at the south pole and f¨/f3 at the
north pole. Call this extension Ωf : S
2 → R and set Ω−f := inf Ωf . We have arrived
at the final result for this section.
Proposition 7.39. The family F 2m : [−`,+`]/∼×T2pi → [−`,+`]/∼×T2pi admits
the expansion
F 2m(u, ψ) =
(
u, ψ + piΩfm
2 + o(m2)
)
. (7.39)
110 7. LOW ENERGY LEVELS OF SYMPLECTIC MAGNETIC FLOWS ON S2
Corollary 7.40. If − f˙
γf3
is not constant, the magnetic flow has infinitely many
periodic orbits on every low energy level. Such condition is satisfied if, for example,
f¨
f3
(0) 6= − f¨
f3
(`) and in this case both F+m and F
−
m are twist maps for small m.
If Ω−f > 0 (namely f˙ = 0 only at the poles and f¨ 6= 0 there), the period T of an
orbit different from the latitudes ζ±m satisfies
T ≥ 2
Ω−f m2
+O
(
1
m3
)
. (7.40)
In particular, ζ+m and ζ
−
m are the only two short orbits.
From the expansion (7.39), it follows that a necessary condition for having ψm
constant for some small m is to require that
− f˙
γf3
= 2k,
for some k ∈ R. We are interested in this condition, because the only way of having
exactly two closed periodic orbits for Xm is to have ψm = am ∈ R/Q. The equation
above can be rewritten as
d
dt
1
f2
= kγ
and integrating in the variable t we find the solution
f =
1√
kΓ + h
, (7.41)
where Γ : [0, `]→ R is a primitive of γ such that Γ(0) = 0, Γ(`) = vol2pi and h ∈ R is
any constant such that h > −k vol2pi , if k < 0,h > 0, if k ≥ 0,
so that the quantity under square root is strictly positive.
A direction of future research would be to study magnetic systems with f given
by (7.41) for suitable choices of γ, k and h, to see if the flow of any of them can be
written down explicitly. In this way one could check if ψm = am ∈ R/Q, for some
value of m.
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