Local rates of Poincaré recurrence for rotations and weak mixing

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

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, 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

Abstract
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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.

Keywords: rotation, Sturmian subshift, linearly recurrent subshift, weak mixing., Poincaré recurrence.

Mathematics Subject Classification: Primary: 37B20; Secondary: 37B1.

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

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