Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems

Abstract

We show that a compound Poisson distribution holds for scaled exceedances of observables $\varphi$ uniquely maximized at a periodic point $\zeta$ in a variety of two-dimensional hyperbolic dynamical systems with singularities $(M,T,\mu )$, including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form $\varphi (z)=-\ln d(z,\zeta )$ where $d$ is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index $\theta$. We calculate $\theta$ in terms of the derivative of the map $T$. Furthermore if we define $M_n=\max \{\varphi ,\dots ,\varphi \circ T^n\}$ and $u_n(\tau )$ by ${\lim }_{n\to \infty }n\mu (\varphi >u_n(\tau ))=\tau$ the maximal process satisfies an extreme value law of form $\mu (M_n\le u_n)=e^{-\theta \tau }$. These results generalize to a broader class of functions maximized at $\zeta$, though the formulas regarding the parameters in the distribution need to be modified.