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March  2003, 9(2): 263-280. doi: 10.3934/dcds.2003.9.263

Pointwise dimensions for Poincaré recurrences associated with maps and special flows

V. Afraimovich 1, , Jean-René Chazottes 2, and  Benoît Saussol 3,

1. 

IICO-UASLP, A. Obregón 64, 78000 San Luis Postosí, SLP, Mexico
2. 

Centre de Physique Théorique, CNRS-Ecole polytechnique, UMR 7644, F-91128 Palaiseau Cedex, France
3. 

CNRS-LAMFA, Université de Picardie Jules Verne, 80039 Amiens, France

Received  November 2001 Revised  May 2002 Published  December 2002

We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
Keywords: special flows., pointwise dimensions, spectra for measures, Poincaré recurrences.
Mathematics Subject Classification: 37C45, 37B20, 37A1.
Citation: V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
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