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Extremes and extremal indices for level set observables on hyperbolic systems. (English) Zbl 1464.37005

This paper focuses primarily on hyperbolic automorphisms of the two-torus, Sinai dispersing billiard maps, and coupled uniformly expanding maps. The first pages introduce all the necessary notation and terminology to understand the paper: extremes for dynamical systems, physical and energy-like observables and stationary stochastic processes.
The main results are stated in Section 2 by Theorems 2.1, 2.5, 2.8 and 2.9. The authors calculate the extreme value distribution, the extremal index and in some cases describe briefly the Poisson return time process. In particular a method for obtaining a nontrivial extremal index which is not due to periodic behavior but rather self-intersection of a set of non-periodic points under the dynamics is described. The proofs of aforementioned theorems are presented in Section 3. Moreover, two main cases with some subcases are considered in this section as below:
Case (a). The line \(\hat{p_{1}}+tv^{+}\), \(-\infty < t < \infty\) contains no point with rational coordinates. This holds for a measure-one set of \(\hat{p_{1}}\) as the set of points in the plane with rational coordinates is countable.
Case(b). The line \(\hat{p_{1}}+tv^{+}\), \(-\infty < t < \infty\) contains a point with rational coordinates. Note that it will contain at most one as the slope of \(v^{+}\) is irrational. Such a point projects to a point \(p_{\mathrm{per}}\) periodic under \(T\).
Case (b1). \(L\) (where \(L\) is a line segment) itself contains \(p_{\mathrm{per}}\), a periodic point.
Furthermore, for hyperbolic automorphisms of the two-torus, the limit laws that arise in these scenarios are established. More generally, it is natural to consider observables whose extremal set \(S\) is no longer (strictly) transverse to the global stable/unstable manifolds, i.e., there exist points of tangency between \(S\) and the global manifolds. Finally, the discussion towards more general observables and nonuniformly hyperbolic systems is considered in Section 4.

MSC:

37A50 Dynamical systems and their relations with probability theory and stochastic processes
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
60G70 Extreme value theory; extremal stochastic processes
60G10 Stationary stochastic processes

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