Abstract
We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measures on attractors.
Highlights
Poincare’s recurrence theorem [22] established that for measure preserving maps points return to neighbourhoods arbitrarily often almost surely
The return time for the first return was quantified by Kac [17] in 1947 and since there have been efforts to describe the return statistics for shrinking neighbourhoods
For a generating partition the natural neighbourhoods are cylinder sets and for those the limiting distributions for entry and return times were shown under various mixing conditions to be exponential with parameter 1
Summary
Poincare’s recurrence theorem [22] established that for measure preserving maps points return to neighbourhoods arbitrarily often almost surely. For a generating partition the natural neighbourhoods are cylinder sets and for those the limiting distributions for entry and return times were shown under various mixing conditions to be exponential with parameter 1 (see for instance [15, 11, 5, 1]). Kupsa constructed an example which has a limiting hitting time distribution almost everywhere and which is not the exponential distribution with parameter 1 For multiple returns it has been established under various mixing conditions that the limiting distribution is Poissonian almost surely. For the return times to metric balls Bρ on manifolds [6] proves the limiting distribution to be Poissonian for the SRB measure on a one-dimensional attractor which allows the construction of a Young tower [24, 25] with exponentially decaying correlations. We use and adapt an argument from [6] Lemma 4.1
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