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February  2005, 13(2): 399-411. doi: 10.3934/dcds.2005.13.399

Fluctuations of the nth return time for Axiom A diffeomorphisms

Jean-René Chazottes 1, and  Renaud Leplaideur 2,

1. 

Centre de Physique Théorique, CNRS-Ecole polytechnique, UMR 7644, F-91128 Palaiseau Cedex
2. 

Laboratoire de Mathématiques, UBO, 6, rue Victor Le Gorgeu, BP 809, F-29285 Brest Cedex, France

Received  June 2004 Revised  December 2004 Published  April 2005

We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition for an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure. As a by-product, we derive precise large deviation estimates and a central limit theorem for these return times. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
Keywords: large deviations, Successive return times, Axiom A, central limit theorem., transfer operator.
Mathematics Subject Classification: 37B20, 37D20, 30C30, 60F05, 60F1.
Citation: Jean-René Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399
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