Measure of maximal entropy for finite horizon Sinai billiard flows

TL;DR

This paper constructs the unique measure of maximal entropy for finite horizon Sinai billiard flows, demonstrating it is Bernoulli under certain entropy conditions, using advanced techniques from dynamical systems theory.

Contribution

It introduces a novel method combining recent equilibrium state results and a leapfrogging approach to establish maximal entropy measures for Sinai billiard flows.

Findings

Existence and uniqueness of the measure of maximal entropy for the flows.

The measure is Bernoulli, indicating strong statistical properties.

The entropy bound applies broadly, likely in generic cases.

Abstract

Using recent work of Carrand on equilibrium states for the billiard map, and bootstrapping via a "leapfrogging" method from a previous article of Baladi and Demers, we construct the unique measure of maximal entropy for two-dimensional finite horizon Sinai (dispersive) billiard flows (and show it is Bernoulli), assuming that the topological entropy of the flow is strictly larger than s_0 log 2 where 0<s_0<1 quantifies the recurrence to singularities. This bound holds in many examples (it is expected to hold generically).