Statistical properties of diffeomorphisms with weak invariant manifolds

José F.  Alves; Davide  Azevedo

doi:10.3934/dcds.2016.36.1

Article Contents

2016, Volume 36, Issue 1: 1-41. Doi: 10.3934/dcds.2016.36.1

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Statistical properties of diffeomorphisms with weak invariant manifolds

José F. Alves^1, and
Davide Azevedo^1,

1.
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto

Received: July 31, 2014

Revised: February 28, 2015

Published: May 2017

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Abstract

We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structure, consisting of a reference set $\Lambda$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial contraction on stable leaves, polynomial backward contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation. We establish a control on the decay of correlations and large deviations of the physical measure of the dynamical system, based on a polynomial control on the Lebesgue measure of the tail of return times. Finally, we present an example of a dynamical system defined on the torus and prove that it verifies the properties of the Gibbs-Markov-Young structure that we considered.

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Mathematics Subject Classification: Primary: 37A05, 37C40, 37D25; Secondary: 60F05, 60F10.

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