Thermodynamic formalism for dispersing billiards

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 $\lim_{t\to 0} P(t)=P(0)$.