Thermodynamic formalism for dispersing billiards

Author:

Baladi Viviane1,Demers Mark F.2

Affiliation:

1. Laboratoire de Probabilités, Statistique et Modélisation (LPSM), CNRS, Sorbonne Université, Université Paris Cité, 4 Place Jussieu, 75005 Paris, France

2. Department of Mathematics, Fairfield University, Fairfield, CT 06824, USA

Abstract

<p style='text-indent:20px;'>For any finite horizon Sinai billiard map <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> on the two-torus, we find <inline-formula><tex-math id="M2">\begin{document}$ t_*&gt;1 $\end{document}</tex-math></inline-formula> such that for each <inline-formula><tex-math id="M3">\begin{document}$ t\in (0,t_*) $\end{document}</tex-math></inline-formula> there exists a unique equilibrium state <inline-formula><tex-math id="M4">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ - t\log J^uT $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M6">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ T $\end{document}</tex-math></inline-formula>-adapted. (In particular, the SRB measure is the unique equilibrium state for <inline-formula><tex-math id="M8">\begin{document}$ - \log J^uT $\end{document}</tex-math></inline-formula>.) We show that <inline-formula><tex-math id="M9">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is exponentially mixing for Hölder observables, and the pressure function <inline-formula><tex-math id="M10">\begin{document}$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $\end{document}</tex-math></inline-formula> is analytic on <inline-formula><tex-math id="M11">\begin{document}$ (0,t_*) $\end{document}</tex-math></inline-formula>. In addition, <inline-formula><tex-math id="M12">\begin{document}$ P(t) $\end{document}</tex-math></inline-formula> is strictly convex if and only if <inline-formula><tex-math id="M13">\begin{document}$ \log J^uT $\end{document}</tex-math></inline-formula> is not <inline-formula><tex-math id="M14">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula>-a.e. cohomologous to a constant, while, if there exist <inline-formula><tex-math id="M15">\begin{document}$ t_a\ne t_b $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ \mu_{t_a} = \mu_{t_b} $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M17">\begin{document}$ P(t) $\end{document}</tex-math></inline-formula> is affine on <inline-formula><tex-math id="M18">\begin{document}$ (0,t_*) $\end{document}</tex-math></inline-formula>. An additional sparse recurrence condition gives <inline-formula><tex-math id="M19">\begin{document}$ \lim_{t\downarrow 0} P(t) = P(0) $\end{document}</tex-math></inline-formula>.</p>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Algebra and Number Theory,Analysis,Applied Mathematics,Algebra and Number Theory,Analysis

Reference26 articles.

1. V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, a Functional Approach, Springer Ergebnisse, 68, 2018.

2. V. Baladi, M. F. Demers.On the measure of maximal entropy for finite horizon Sinai billiard maps, J. Amer. Math. Soc., 33 (2020), 381-449.

3. V. Baladi, M. F. Demers, C. Liverani.Exponential decay of correlations for finite horizon Sinai billiard flows, Inventiones Math., 211 (2018), 39-177.

4. P. Bálint, J. De Simoi and I. P. Tóth, A proof of Theorem 5.67 in "Chaotic Billiards" by Chernov and Markarian, preprint, available from: http://www.math.utoronto.ca/jacopods/pdf/kolya-patch.pdf.

5. C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encycl. Math. Sciences, 102, Springer, Berlin, 2005.

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