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Explicit coupling argument for non-uniformly hyperbolic transformations

Part of: Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  06 July 2018

A. Korepanov ,
Z. Kosloff  and
I. Melbourne
A. Korepanov
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (a.korepanov@warwick.ac.uk; i.melbourne@warwick.ac.uk)
Z. Kosloff
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (a.korepanov@warwick.ac.uk; i.melbourne@warwick.ac.uk)
I. Melbourne
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (a.korepanov@warwick.ac.uk; i.melbourne@warwick.ac.uk)
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Abstract

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

Keywords

couplingmixingdecay of correlationsYoung towersexplicit rates

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 1 , February 2019 , pp. 101 - 130
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

*

Permanent address: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel (zemer.kosloff@mail.huji.ac.il).