# American Institute of Mathematical Sciences

January  2005, 12(1): 175-183. doi: 10.3934/dcds.2005.12.175

## Local rates of Poincaré recurrence for rotations and weak mixing

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

Received  March 2003 Revised  June 2004 Published  December 2004

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.
Citation: Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
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