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

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.