## Abstract

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: Download PDF file or
e-mail: mikefield@gmail.com or Michael.J.Field@rice.edu

Professor Mike Field

Department of Mathematics

Rice University

6100 S Main St

Houston

TX 77005-1892