Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems. (English) Zbl 1385.37049
The topic of the article is the extreme value law for two-dimensional hyperbolic systems with singularities (such as defined in [A. Katok and 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 [J. M. Freitas et al., Nonlinearity 27, No. 7, 1669–1687 (2014; Zbl 1348.37061)] to the case of periodic points.
Reviewer: Ivan Podvigin (Novosibirsk)
MSC:
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 60G70 | Extreme value theory; extremal stochastic processes |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 37A25 | Ergodicity, mixing, rates of mixing |
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