# Ergodic Theory of Equivariant Diffeomorphisms: Markov Partitions and Stable Ergodicity

Share this page
*Michael Field; Matthew Nicol*

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 \(M\).

#### Table of Contents

# Table of Contents

## Ergodic Theory of Equivariant Diffeomorphisms: Markov Partitions and Stable Ergodicity

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Equivariant Geometry and Dynamics 918 free
- Chapter 3. Technical preliminaries 1928
- Part 1. Markov partitions 2938
- Chapter 4. Markov partitions for finite group actions 3140
- Chapter 5. Transversally hyperbolic sets 4756
- 5.1. Transverse hyperbolicity 4756
- 5.2. Properties of transversally hyperbolic sets 5059
- 5.3. Γ-expansiveness 5261
- 5.4. Stability properties of transversally hyperbolic sets 5362
- 5.5. Subshifts of finite type and attractors 5463
- 5.6. Local product structure 5564
- 5.7. Expansiveness and shadowing 5665
- 5.8. Stability of basic sets 5867

- Chapter 6. Markov partitions for basic sets 5968

- Part 2. Stable Ergodicity 7180
- Appendix A. On the absolute continuity of v 93102
- Bibliography 97106