Search for: All records

In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate.

 

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

 

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.