A family of natural equilibrium measures for Sinai billiard flows

TL;DR

This paper establishes a rigorous connection between equilibrium measures of Sinai billiard flows and their associated maps, proving existence, uniqueness, and Bernoulli properties for a broad class of potentials.

Contribution

It introduces a new definition of topological pressure for Sinai billiard maps and proves the existence and uniqueness of equilibrium states with Bernoulli properties.

Findings

Equilibrium states are unique and Bernoulli for a broad class of potentials.

The flow invariant measure is Bernoulli and flow adapted.

Examples of billiard tables with open sets of suitable potentials are provided.

Abstract

The Sinai billiard flow on the two-torus, i.e., the periodic Lorentz gas, is a continuous flow, but it is not everywhere differentiable. Assuming finite horizon, we relate the equilibrium states of the flow with those of the Sinai billiard map $T$ -- which is a discontinuous map. We propose a definition for the topological pressure $P_*(T,g)$ associated to a potential $g$. We prove that for any piecewise H\"older potential $g$ satisfying a mild assumption, $P_*(T,g)$ is equal to the definitions of Bowen using spanning or separating sets. We give sufficient conditions under which a potential gives rise to equilibrium states for the Sinai billiard map. We prove that in this case the equilibrium state $\mu_g$ is unique, Bernoulli, adapted and gives positive measure to all nonempty open sets. For this, we make use of a well chosen transfer operator acting on anisotropic Banach spaces, and construct the measure by pairing its maximal eigenvectors. Last, we prove that the flow invariant probability measure $\bar \mu_g$, obtained by taking the product of $\mu_g$ with the Lebesgue measure along orbits, is Bernoulli and flow adapted. We give examples of billiard tables for which there exists an open set of potentials satisfying those sufficient conditions.