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Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Part of: Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  04 June 2021

WAEL BAHSOUN  and
CARLANGELO LIVERANI
WAEL BAHSOUN*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK
CARLANGELO LIVERANI
Affiliation:
Dipartimento di Matematica, Università di Roma II, Tor Vergata, Via della Ricerca Scientifica, 00133Roma, Italy (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

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Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

Keywords

anisotropic Banach spacesAnosov diffeomorphsimsfoliation regularitytransfer operators

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2431 - 2467
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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