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Thermodynamic formalism for dispersing billiards

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}.

Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10382743
Journal Name:
Journal of Modern Dynamics
Volume:
18
Issue:
0
Page Range or eLocation-ID:
415
ISSN:
1930-5311
Sponsoring Org:
National Science Foundation
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