Local rates of Poincaré recurrence for rotations and weak mixing

Jean René Chazottes; F. Durand

doi:10.3934/dcds.2005.12.175

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2005, Volume 12, Issue 1: 175-183. Doi: 10.3934/dcds.2005.12.175

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Local rates of Poincaré recurrence for rotations and weak mixing

Jean René Chazottes^1, and
F. Durand^2,

1.
CNRS-CPHT, École Polytechnique, 91128 Palaiseau Cedex
2.
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex 1

Received: February 28, 2003

Revised: May 31, 2004

Published: March 2005

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Abstract

We study the lower and upper local rates of Poincaré recurrence of rotations on the circle by means of symbolic dynamics. As a consequence, we show that if the lower rate of Poincaré recurrence of an ergodic dynamical system $(X,\mathcal F, \mu, T)$ is greater or equal to 1 $\mu$-almost everywhere, then it is weakly mixing.

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Mathematics Subject Classification: Primary: 37B20; Secondary: 37B10.

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