On the measure of maximal entropy for finite horizon Sinai Billiard maps

Abstract

The Sinai billiard map T T on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition h ∗ h_* for the topological entropy of T T . We prove that h ∗ h_* is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure μ ∗ \mu _* of maximal entropy for T T (i.e., h μ ∗ ( T ) = h ∗ h_{\mu _*}(T)=h_* ), we show that μ ∗ \mu _* has full support and is Bernoulli, and we prove that μ ∗ \mu _* is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to h ∗ h_* . Second, h ∗ h_* is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction of T T to a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map T T has at least C e n h ∗ C e^{nh_*} periodic points of period n n for all n ∈ N n \in \mathbb {N} .