Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

Authors
Breuillard, E 
Varjú, PP 

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Article
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Abstract

The exponential growth rate of non polynomially growing subgroups of GLd is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semigroup generated by the maps x↦λ⋅x±1, where λ is an algebraic number. We give a bound in terms of the Mahler measure of λ. We also derive a bound on the dimension of Bernoulli convolutions.

Publication Date
2020
Online Publication Date
2020-04-20
Acceptance Date
2018-06-10
Keywords
math.CA, math.CA, math.GR, math.PR
Journal Title
Journal d'Analyse Mathematique
Journal ISSN
0021-7670
1565-8538
Volume Title
140
Publisher
Springer Science and Business Media LLC
Sponsorship
Royal Society (UF140146)
Simons Foundation (LETTER DATED 10-NOV-09)
Simons Foundation Royal Society ERC