Part I:

We obtain basic stability and structural results, including a generalization of Markov partitions,
for equivariant diffeomorphisms which are hyperbolic transverse to a compact Lie group action. We also prove the existence of Markov partitions on the orbits space of a partially hyperbolic invariant set. (We assume all group orbits have the same dimension, equal to that of the center foliation.)

Part II:

Let G be a compact connected Lie group acting smoothly on M. Let F be a smooth G-equivariant diffeomorphism of M such that the restriction of F to the G- and F-invariant set L of M is partially hyperbolic with the center foliation given by G-orbits. We assume that the G-orbits all have dimension equal to that of G but do not require that the action of G on L is free. We show that there is a naturally defined F-invariant measure m of maximal entropy on L. In this setting we prove a version of the Livsic 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,L,m) is stably ergodic if G is semisimple, or G is abelian and the topological dimension of L/G is zero, or L is an attractor. In the case when L is an attractor, we show that L is generically a stably SRB attractor within the class of G-equivariant diffeomorphisms of M.

For preprint, email: mikefield@gmail.com

Professor Mike Field
Department of Mathematics
Imperial College
London SW7 2AZ