A direct proof of the existence of MME for finite horizon Sinai billiards

Abstract

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gaz, is a discontinuous map. Assuming finite horizon and another condition we introduce -- namely \emph{negligible singularities} -- we prove that the metric pressure map associated with the billiard map $T$ is upper semi-continuous, as well as the compactness of the set of $T$-invariant measures. In particular, for the potentials $g\equiv 0$ and $g=-h_{\mathrm{top}}({\Phi }^1)\tau$, we recover the recent results of the existence of measures of maximal entropy (MME) for both the billiard map and flow ${\Phi }^t$, due to Baladi and Demers for $T$, jointly with Carrand for ${\Phi }^t$. We prove that the negligible singularities condition is generic among the billiard table with $C^{3+\alpha }$ boundary, with respect to the $C^{3+\alpha }$ topology. For finite horizon Sinai billiards, we provide bounds on the defect of upper semi-continuity of the metric pressure map and on the topological tail entropy. Assuming that singularities are negligible and sparse recurrence hold, we deduce the equidistribution of the periodic orbits with respect to the unique MME. We provide examples of billiard tables satisfying both conditions.