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 $$\phi $$ϕ uniquely maximized at a periodic point $$\zeta $$ζ in a variety of two-dimensional hyperbolic dynamical systems with singularities $$(M,T,\mu )$$(M,T,μ), including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form $$\phi (z)=-\ln d(z,\zeta )$$ϕ(z)=-lnd(z,ζ) 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 $$\theta $$θ. We calculate $$\theta $$θ in terms of the derivative of the map T. Furthermore if we define $$M_n=\max \{\phi ,\ldots ,\phi \circ T^n\}$$Mn=max{ϕ,…,ϕ∘Tn} and $$u_n (\tau )$$un(τ) by $$\lim _{n\rightarrow \infty } n\mu (\phi >u_n (\tau ) )=\tau $$limn→∞nμ(ϕ>un(τ))=τ the maximal process satisfies an extreme value law of form $$\mu (M_n \le u_n)=e^{-\theta \tau }$$μ(Mn≤un)=e-θτ. These results generalize to a broader class of functions maximized at $$\zeta $$ζ, though the formulas regarding the parameters in the distribution need to be modified.