Fluctuations of the  nth return time for Axiom A diffeomorphisms

Jean-René Chazottes; Renaud Leplaideur

doi:10.3934/dcds.2005.13.399

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2005, Volume 13, Issue 2: 399-411. Doi: 10.3934/dcds.2005.13.399

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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

Received: May 31, 2004

Revised: November 30, 2004

Published: January 2005

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Abstract

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.

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Mathematics Subject Classification: 37B20, 37D20, 30C30, 60F05, 60F10.

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