On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps

Abstract

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 non grazing 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 non-compact 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 large enough $n \in \mathbb{N}$.