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We show that a compound Poisson distribution holds for scaled exceedances ofobservables $\phi$ uniquely maximized at a periodic point $\zeta$ in a varietyof two-dimensional hyperbolic dynamical systems with singularities $(M,T,\mu)$,including the billiard maps of Sinai dispersing billiards in both the finiteand infinite horizon case. The observable we consider is of form $\phi (z)=-\lnd(z,\zeta)$ where $d$ is a metric defined in terms of the stable and unstablefoliation. The compound Poisson process we obtain is a P\'olya-Aepplidistibution of index $\theta$. We calculate $\theta$ in terms of the derivativeof the map $T$. Furthermore if we define $M_n=\max\{\phi,\ldots,\phi\circT^n\}$ 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 $\mu (M_n \leu_n)=e^{-\theta \tau}$. These results generalize to a broader class offunctions maximized at $\zeta$, though the formulas regarding the parameters inthe distribution need to be modified.

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