Thermodynamic formalism for dispersing billiards

Abstract

For any finite horizon Sinai billiard map \begin{document}$ T $\end{document} on the two-torus, we find \begin{document}$ t_*>1 $\end{document} such that for each \begin{document}$ t\in (0,t_*) $\end{document} there exists a unique equilibrium state \begin{document}$ \mu_t $\end{document} for \begin{document}$ - t\log J^uT $\end{document}, and \begin{document}$ \mu_t $\end{document} is \begin{document}$ T $\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$ - \log J^uT $\end{document}.) We show that \begin{document}$ \mu_t $\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $\end{document} is analytic on \begin{document}$ (0,t_*) $\end{document}. In addition, \begin{document}$ P(t) $\end{document} is strictly convex if and only if \begin{document}$ \log J^uT $\end{document} is not \begin{document}$ \mu_t $\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$ t_a\ne t_b $\end{document} with \begin{document}$ \mu_{t_a} = \mu_{t_b} $\end{document}, then \begin{document}$ P(t) $\end{document} is affine on \begin{document}$ (0,t_*) $\end{document}. An additional sparse recurrence condition gives \begin{document}$ \lim_{t\downarrow 0} P(t) = P(0) $\end{document}.