Topological entropy and pressure for finite-horizon Sinai billiards

Abstract

This brief survey describes recent progress in our understanding of a variety of equilibrium states for finite-horizon dispersing billiard maps in two dimensions. In particular, we review formulations of topological entropy and pressure for the family of geometric potentials $-t\log J^{u}T$, where $J^{u}T$ denotes the unstable Jacobian of the map and $t\in \mathbb{R}$. We summarize recent results, proving the existence and uniqueness of related equilibrium states for some range of $t\ge 0$, including $[0,1]$. In this family, $t=0$ corresponds to the measure of maximal entropy, while $t=1$ corresponds to the smooth invariant measure for the billiard map. In addition, variational principles are presented which express topological notions of pressure and entropy as the supremum of their measure-theoretic counterparts.