  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.
