Every finite horizon Sinai billiard map has a unique measure of maximal entropy

TL;DR

This paper proves that all finite horizon Sinai billiard maps have a unique measure of maximal entropy, extending previous results without requiring the sparse recurrence condition, using a concrete construction.

Contribution

It provides a new proof establishing the existence and uniqueness of the measure of maximal entropy for all finite horizon Sinai billiard maps, removing the sparse recurrence assumption.

Findings

Confirmed the uniqueness of the measure of maximal entropy for all finite horizon Sinai billiard maps.

Developed a concrete construction of the measure as a product of Hausdorff measures on associated subshifts.

Extended previous results to a broader class of Sinai billiard systems.

Abstract

Finite horizon Sinai billiard maps are examples of uniformly hyperbolic systems with singularities. These discontinuities make it more difficult to develop the classical theory of thermodynamic formalism. Nevertheless, Baladi and Demers established a variational principle for these systems, and proved that if the billiard table satisfies a certain sparse recurrence condition, then there is a unique measure of maximal entropy. We extend this existence and uniqueness result to all finite horizon Sinai billiard maps by giving a new proof that does not rely on the sparse recurrence condition. Our construction is very concrete: the unique MME is obtained as the product of the Hausdorff measures on the one-sided subshifts associated to the billiard map.