A direct proof of the existence of MME for finite horizon Sinai billiards
The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. We provide two separate conditions for a finite horizon billiard to admit equilibrium states. The first one -- namely \emph{negligible singularities} -- only concerns the billiard table and implies upper semi-continuity for the metric pressure map of $T$. The condition of negligible singularities is shown to be generic (dense $G_δ$) among billiard tables with $C^{3+α}$ boundaries, in the $C^{3+α}$ topology, and implies the sparse recurrence condition for the null potential. Compactness of the set of $T$-invariant measures is proved. For finite horizon Sinai billiards, we provide a bound on the defect of upper semi-continuity of the metric pressure map. From it, a condition of existence of equilibrium states only involving potentials is deduced. In both situations, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(Φ^1) τ$, we recover and extend the recent results of the existence of measures of maximal entropy (MME) for both the map $T$ and the flow $Φ^t$, due to Baladi and Demers for $T$, jointly with Carrand for $Φ^t$. Upper semi-continuity of the metric entropy is shown. Assuming negligible singularities, equidistribution of the periodic orbits with respect to the unique MME is proved. A bound on the topological tail entropy is given.