Thermodynamic formalism for dispersing billiards

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>