Thermodynamic formalism for dispersing billiards

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 $\mu_t$ $- t\log J^uT$, and is T-adapted. (In particular, SRB measure \log J^uT$.) We show exponentially mixing Holder observables, pressure function $P(t)=\sup_\mu \{h_\mu -\int J^uT d \mu\}$ analytic (0,t_*). In addition, P(t) strictly convex if only $\log J^uT$ not a.e. cohomologous to constant, while, exist $t_a\ne t_b$ with $\mu_{t_a}= \mu_{t_b}$, then affine An additional sparse recurrence condition gives $\lim_{t\to 0} P(t)=P(0)$.