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

Viviane Baladi, Mark Demers

Published 2018-07-06Version 1

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 compatible with the definitions of Bowen using spanning or separating sets. If $h_* \ge \log 2$ (our actual condition is weaker), then 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 different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to $h_*$. Second, $h_*$ is compatible with 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, the map $T$ has at least $C e^{pnh_*}$ periodic points of period $pn$ for all $n$ and some $p\ge 1$.