Return times distribution for Markov towers with decay of correlations

Details Return times distribution for Markov towers with decay of correlations Return times distribution for Markov towers with decay of correlations

2010-10-05

arxiv.org/abs/1010.1029v1

Mathematics

Dynamical Systems

Scientific paper

In this paper we prove two results. First we show that dynamical systems with an $\alpha$-mixing have in the limit Poisson distributed return times almost everywhere. We use the Chen-Stein method to also obtain rates of convergence. Our theorem improves on previous results by allowing for infinite partitions and dropping the requirement that the invariant measure have finite entropy with respect to the given partition. As has been shown elsewhere, the limiting distribution at periodic points is not Poissonian (but compound Poissonian), therefore our result applies to cylinder neighbourhoods that don't exhibit local periodic behaviour. In the second part we prove that Lai-Sang Young's Markov Towers have Poisson distributed return times if the correlations decay for observables that are H\"older continous and $L^\infty$ bounded.

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