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In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate.

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}.