## Abstract

Let G be a compact connected Lie group, and M be a compact G-manifold. Let F be a smooth
G-equivariant diffeomorphism of M, and X be a compact G- and F-invariant subset of M.
We assume that X is partially hyperbolic, with central foliation given by G-orbits.
Let f:X/G -> X/G denote the homeomorphism induced by F on the orbit space.
Subject to certain conditions, we show that the set of topologically transitive Holder
(or Ck) equivariant homeomorphisms of X covering f is open and dense in
Holder (Ck) topology. Our results apply to skew and principal extensions by a compact
connected semisimple Lie group over a general basic hyperbolic set.

For preprint,
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