For any finite horizon Sinai billiard map
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on the two-torus, we find\begin{document}$ T $\end{document} such that for each\begin{document}$ t_*>1 $\end{document} there exists a unique equilibrium state\begin{document}$ t\in (0,t_*) $\end{document} for\begin{document}$ \mu_t $\end{document} , and\begin{document}$ - t\log J^uT $\end{document} is\begin{document}$ \mu_t $\end{document} -adapted. (In particular, the SRB measure is the unique equilibrium state for\begin{document}$ T $\end{document} .) We show that\begin{document}$ - \log J^uT $\end{document} is exponentially mixing for Hölder observables, and the pressure function\begin{document}$ \mu_t $\end{document} is analytic on\begin{document}$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $\end{document} . In addition,\begin{document}$ (0,t_*) $\end{document} is strictly convex if and only if\begin{document}$ P(t) $\end{document} is not\begin{document}$ \log J^uT $\end{document} -a.e. cohomologous to a constant, while, if there exist\begin{document}$ \mu_t $\end{document} with\begin{document}$ t_a\ne t_b $\end{document} , then\begin{document}$ \mu_{t_a} = \mu_{t_b} $\end{document} is affine on\begin{document}$ P(t) $\end{document} . An additional sparse recurrence condition gives\begin{document}$ (0,t_*) $\end{document} .\begin{document}$ \lim_{t\downarrow 0} P(t) = P(0) $\end{document} -
null (Ed.)We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.more » « less