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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the measure of maximal entropy for finite horizon Sinai Billiard maps
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by Viviane Baladi and Mark F. Demers HTML | PDF
J. Amer. Math. Soc. 33 (2020), 381-449 Request permission


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:
  • Mark F. Demers
  • Affiliation: Department of Mathematics, Fairfield University, Fairfield, Connecticut 06824
  • MR Author ID: 763971
  • Email:
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 33 (2020), 381-449
  • MSC (2010): Primary 37D50; Secondary 37C30, 37B40, 37A25, 46E35, 47B38
  • DOI:
  • MathSciNet review: 4073865