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On the measure of maximal entropy for finite horizon Sinai Billiard maps


Authors: Viviane Baladi and Mark F. Demers
Journal: J. Amer. Math. Soc. 33 (2020), 381-449
MSC (2010): Primary 37D50; Secondary 37C30, 37B40, 37A25, 46E35, 47B38
DOI: https://doi.org/10.1090/jams/939
Published electronically: January 6, 2020
MathSciNet review: 4073865
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Abstract: The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure $\mu _*$ of maximal entropy for $T$ (i.e., $h_{\mu _*}(T)=h_*$), we show that $\mu _*$ has full support and is Bernoulli, and we prove that $\mu _*$ is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to $h_*$. Second, $h_*$ is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction of $T$ to a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map $T$ has at least $C e^{nh_*}$ periodic points of period $n$ for all $n \in \mathbb {N}$.


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Additional Information

Viviane Baladi
Affiliation: CNRS, Institut de Mathématiques de Jussieu (IMJ-PRG), Sorbonne Université, 4, Place Jussieu, 75005 Paris, France
Address at time of publication: Laboratoire de Probabilités, Statistique et Modélisation (LPSM), CNRS, Sorbonne Université, Université de Paris, 4, Place Jussieu, 75005 Paris, France
MR Author ID: 29810
Email: baladi@lpsm.paris

Mark F. Demers
Affiliation: Department of Mathematics, Fairfield University, Fairfield, Connecticut 06824
MR Author ID: 763971
Email: mdemers@fairfield.edu

Received by editor(s): August 25, 2018
Received by editor(s) in revised form: August 19, 2019
Published electronically: January 6, 2020
Additional Notes: The first author’s research was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 787304).
The second author was partly supported by NSF grants DMS 1362420 and DMS 1800321.
Article copyright: © Copyright 2020 American Mathematical Society