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Quantitative Pesin theory for Anosov diffeomorphisms and flows

Published online by Cambridge University Press:  22 May 2017

SÉBASTIEN GOUËZEL  and
LUCHEZAR STOYANOV
SÉBASTIEN GOUËZEL
Affiliation:
Laboratoire Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes, France email sebastien.gouezel@univ-nantes.fr
LUCHEZAR STOYANOV
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley 6009 WA, Australia email luchezar.stoyanov@uwa.edu.au
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Abstract

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 1 , January 2019 , pp. 159 - 200
Copyright
© Cambridge University Press, 2017 

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