Entropy of Bernoulli convolutions and uniform exponential growth for linear groups
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Authors
Breuillard, Emmanuel
Varjú, PP
Publication Date
2020-03Journal Title
Journal d'Analyse Mathematique
ISSN
0021-7670
Publisher
Springer Science and Business Media LLC
Volume
140
Issue
2
Pages
443-481
Type
Article
This Version
AM
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Breuillard, E., & Varjú, P. (2020). Entropy of Bernoulli convolutions and uniform exponential growth for linear groups. Journal d'Analyse Mathematique, 140 (2), 443-481. https://doi.org/10.1007/s11854-020-0100-0
Abstract
The exponential growth rate of non polynomially growing subgroups of $GL_d$
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 $x\mapsto
\lambda\cdot x\pm 1$, where $\lambda$ is an algebraic number. We give a bound
in terms of the Mahler measure of $\lambda$. We also derive a bound on the
dimension of Bernoulli convolutions.
Sponsorship
Simons Foundation
Royal Society
ERC
Funder references
Royal Society (UF140146)
Simons Foundation (LETTER DATED 10-NOV-09)
Identifiers
External DOI: https://doi.org/10.1007/s11854-020-0100-0
This record's URL: https://www.repository.cam.ac.uk/handle/1810/287816
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