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 $-t\log J^{u}T$, and ${\mu }_t$ is T-adapted. (In particular, the SRB measure is the unique equilibrium state for $-\log J^{u}T$.) We show that ${\mu }_t$ is exponentially mixing for Holder observables, and the pressure function $P(t)={\sup }_{\mu }\{h_{\mu }-\int t\log J^{u}Td\mu \}$ is analytic on (0,t_*). In addition, P(t) is strictly convex if and only if $\log J^{u}T$ 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 ${\lim }_{t\to 0}P(t)=P(0)$.

Version v3 is the electronic copy of the published version in Journal of Modern Dynamics