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Entry Times Distribution for Mixing Systems

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Abstract

We consider the return times dynamics to Bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties. These mixing properties are satisfied for instance by systems that allow Young tower constructions. We show that the higher order return times to Bowen balls are in the limit Poisson distributed. We also provide a general result for the asymptotic behavior of the recurrence time for Bowen balls for ergodic systems and those with specification.

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Notes

  1. f can be computed recursively from the Stein equation:

    $$\begin{aligned} f(k)=\frac{(k-1)!}{t^k}\sum _{i=0}^{k-1} \left( h(i)-\mu _0(h)\right) \frac{t^i}{i!} =-\frac{(k-1)!}{t^k}\sum _{i=k}^{\infty } \left( h(i)-\mu _0(h)\right) \frac{t^i}{i!} , \quad \text {} \forall k\in \mathbb {N}. \end{aligned}$$

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089-2532, USA

    N. Haydn & F. Yang

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  1. N. Haydn
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  2. F. Yang
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Correspondence to N. Haydn.

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Haydn, N., Yang, F. Entry Times Distribution for Mixing Systems. J Stat Phys 163, 374–392 (2016). https://doi.org/10.1007/s10955-016-1487-y

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