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

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.