Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems (Q1696960)

The topic of the article is the extreme value law for two-dimensional hyperbolic systems with singularities (such as defined in [\textit{A. Katok} and \textit{J.-M. Strelcyn}, Invariant manifolds, entropy and billiards; smooth maps with singularities. Berlin etc.: Springer-Verlag (1986; Zbl 0658.58001)]) including Sinai dispersing billiards with finite and infinite horizon as well. The observable that the authors consider is of form \({\phi(z)=-\ln d(z,x)}\), where \(d\) is a dynamically adapted metric (i.e., defined in terms of the stable and unstable foliation) and \(x\) is a periodic point of transformation \(T\) (with period \(q\)). In such case a geometric distribution is obtained with parameter equal to the extremal index: \[ {\theta=1-\frac{1}{|DT^q_u(x)|}}, \] where \({DT^q_u(x)}\) is the derivative of \(T^q\) in the unstable direction at \(x\). This result extends the findings in [\textit{J. M. Freitas} et al., Nonlinearity 27, No. 7, 1669--1687 (2014; Zbl 1348.37061)] to the case of periodic points.