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Ergodic Theory of Equivariant Diffeomorphisms: Markov Partitions and Stable Ergodicity
Michael Field, University of Houston, TX, and Matthew Nicol, University of Surrey, Guildford, England
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Memoirs of the American Mathematical Society
2004; 100 pp; softcover
Volume: 169
ISBN-10: 0-8218-3599-8
ISBN-13: 978-0-8218-3599-9
List Price: US$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/169/803
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We obtain stability and structural results for equivariant diffeomorphisms which are hyperbolic transverse to a compact (connected or finite) Lie group action and construct `\(\Gamma\)-regular' Markov partitions which give symbolic dynamics on the orbit space. We apply these results to the situation where \(\Gamma\) is a compact connected Lie group acting smoothly on \(M\) and \(F\) is a smooth (at least \(C^2\)) \(\Gamma\)-equivariant diffeomorphism of \(M\) such that the restriction of \(F\) to the \(\Gamma\)- and \(F\)-invariant set \(\Lambda\subset M\) is partially hyperbolic with center foliation given by \(\Gamma\)-orbits. On the assumption that the \(\Gamma\)-orbits all have dimension equal to that of \(\Gamma\), we show that there is a naturally defined \(F\)- and \(\Gamma\)-invariant measure \(\nu\) of maximal entropy on \(\Lambda\) (it is not assumed that the action of \(\Gamma\) is free). In this setting we prove a version of the Livšic regularity theorem and extend results of Brin on the structure of the ergodic components of compact group extensions of Anosov diffeomorphisms. We show as our main result that generically \((F,\Lambda,\nu)\) is stably ergodic (openness in the \(C^2\)-topology). In the case when \(\Lambda\) is an attractor, we show that \(\Lambda\) is generically a stably SRB attractor within the class of \(\Gamma\)-equivariant diffeomorphisms of \(M\).

Readership

Graduate students and research mathematicians interested in dynamical systems and ergodic theory.

Table of Contents

  • Introduction
  • Equivariant geometry and dynamics
  • Technical preliminaries
Part 1. Markov Partitions
  • Markov partitions for finite group actions
  • Transversally hyperbolic sets
  • Markov partitions for basic sets
Part 2. Stable Ergodicity
  • Preliminaries
  • Livšic regularity and ergodic components
  • Stable ergodicity
  • Appendix A. On the absolute continuity of \(\nu\)
  • Appendix. Bibliography
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