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Statistics of closest return for some non-uniformly hyperbolic systems. (English) Zbl 1002.37019

The author considers \(C^{2}\)-maps on the interval satisfying some technical assumptions (however, it is known that many interesting interval maps satisfy these technical assumptions). An ergodic invariant probability measure \(\mu\) is constructed. Given \(x\) the random variable \(Z_{n}^{x}\) is defined as \(Z_{n}^{x}(y):= \sup_{j=0,1,\dots ,n}(-\log|x- T^{j}y|)\) (the “closest return” to \(x\)). It is proved that for Lebesgue-almost all \(x\) one has \(\lim_{n\to\infty} \mu (Z_{n}^{x}<v+\log n)= e^{-2h(x)e^{-v}}\) for all \(v\in{\mathbb R}\), where \(h\) denotes the density of \(\mu\). A similar result is obtained for the closest return of the orbit of \(x\) to \(x\).
Reviewer: Peter Raith (Wien)

MSC:

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
34C40 Ordinary differential equations and systems on manifolds
37A05 Dynamical aspects of measure-preserving transformations
37E05 Dynamical systems involving maps of the interval
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