Thermodynamic formalism for dispersing billiards

TL;DR

This paper studies the thermodynamic formalism for Sinai billiard maps, establishing existence, uniqueness, and properties of equilibrium states for certain potentials, and analyzing the analyticity and convexity of the pressure function.

Contribution

It introduces a parameter range for which unique equilibrium states exist and characterizes their properties, including mixing and pressure function behavior, for dispersing billiards.

Findings

Existence of a parameter t_* where equilibrium states are unique.

Analyticity and strict convexity of the pressure function P(t).

Exponential mixing of the equilibrium states.

Abstract

For any finite horizon Sinai billiard map T on the two-torus, we find t_*>1 such that for each t in (0,t_*) there exists a unique equilibrium state $\mu_t$ for $- t\log J^uT$, and $\mu_t$ is T-adapted. (In particular, the SRB measure is the unique equilibrium state for $- \log J^uT$.) We show that $\mu_t$ is exponentially mixing for Holder observables, and the pressure function $P(t)=\sup_\mu \{h_\mu -\int t\log J^uT d \mu\}$ is analytic on (0,t_*). In addition, P(t) is strictly convex if and only if $\log J^uT$ is not $\mu_t$ a.e. cohomologous to a constant, while, if there exist $t_a\ne t_b$ with $\mu_{t_a}= \mu_{t_b}$, then P(t) is affine on (0,t_*). An additional sparse recurrence condition gives $\lim_{t\to 0} P(t)=P(0)$.