Recurrence for random dynamical systems

Philippe Marie; Jérôme Rousseau

doi:10.3934/dcds.2011.30.1

Article Contents

2011, Volume 30, Issue 1: 1-16. Doi: 10.3934/dcds.2011.30.1

This issue Previous Article Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations Next Article SRB measures for certain Markov processes

Recurrence for random dynamical systems

Philippe Marie^1, and
Jérôme Rousseau^2,

1.
Université d'Aix-Marseille, Centre de physique théorique, UMR 6206 CNRS, Campus de Luminy, Case 907, 13288 Marseille cedex 9 and Université du Sud, Toulon-Var
2.
Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques CNRS UMR 6205,6 avenue Victor le Gorgeu, CS93837, F-29238 Brest Cedex 3

Received: November 30, 2009

Revised: April 30, 2010

Published: December 2010

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Abstract

This paper is a first step in the study of the recurrence behaviour in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.

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Mathematics Subject Classification: Primary: 37H, 37C45, 37B20; Secondary: 37A25.

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