Recurrence rate in rapidly mixing dynamical systems

Benoît Saussol

doi:10.3934/dcds.2006.15.259

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2006, Volume 15, Issue 1: 259-267. Doi: 10.3934/dcds.2006.15.259

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Recurrence rate in rapidly mixing dynamical systems

Benoît Saussol^1,

1.
Département de Mathématiques - CNRS UMR 6205, Université de Bretagne Occidentale, 6 Avenue le Gorgeu, CS 93837, 29238 Brest cedex 3

Received: October 31, 2004

Revised: June 30, 2005

Published: February 2006

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Abstract

For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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