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

TL;DR

This paper demonstrates that in two-dimensional hyperbolic systems with singularities, the scaled exceedances of certain observables follow a compound Poisson distribution, specifically a Pólya-Aeppli distribution, with parameters linked to the system's dynamics.

Contribution

It establishes the compound Poisson law for hitting times to periodic orbits in hyperbolic systems, including billiard maps, and computes the distribution's parameters explicitly.

Findings

Hitting times follow a Pólya-Aeppli distribution with index θ.

Explicit formula for θ in terms of the derivative of T.

Maximal process obeys an extreme value law with parameter θ.

Abstract

We show that a compound Poisson distribution holds for scaled exceedances of observables $\phi$ 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 $\phi (z)=-\ln d(z,\zeta)$ where $d$ is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a P\'olya-Aeppli distibution of index $\theta$. We calculate $\theta$ in terms of the derivative of the map $T$. Furthermore if we define $M_n=\max\{\phi,\ldots,\phi\circ T^n\}$ and $u_n (\tau)$ by $\lim_{n\to \infty} n\mu (\phi >u_n (\tau) )=\tau$ 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.