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.
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