On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps

Abstract

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_{\ast }$ for the topological entropy of $T$. We prove that $h_{\ast }$ 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 ${\mu }_{\ast }$ of maximal entropy for $T$ (i.e., $h_{{\mu }_{\ast }}(T)=h_{\ast }$), we show that ${\mu }_{\ast }$ has full support and is Bernoulli, and we prove that ${\mu }_{\ast }$ is the unique measure of maximal entropy, and that it is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to $h_{\ast }$. Second, $h_{\ast }$ is equal to the Bowen--Pesin--Pitskel topological entropy of the restriction of $T$ to a non-compact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map $T$ has at least $C{e}^{n{h}_{\ast }}$ periodic points of period $n$ for all large enough $n\in \mathbb{N}$.

69 +1 pages, 3 Figures. Version v3 is the electronic copy of the published version. v4 contains a supplement describing minor typos

References in corpus (1)

• Fluctuation of the entropy production for the Lorentz gas under small external forces

Cited by in corpus (7)

• Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows
• Marked Length Spectral determination of analytic chaotic billiards with axial symmetries
• Thermodynamic formalism for dispersing billiards
• Closed geodesics with prescribed intersection numbers
• Projective Cones for Sequential Dispersing Billiards
• State-to-State Cosmology: a new view on the cosmological arrow of time and the past hypothesis
• Anisotropic spaces for the bilateral shift