Stable topological transitivity of skew and principal extensions

Abstract

Let G be a compact connected Lie group, and M be acompactG-manifold. Let Φ be a smooth G-equivariant diffeomorphism ofM, and Λ⊂M be a compact G- and Φ-invariantsubset. We assume that Λ is partially hyperbolic, with centralfoliation given by G-orbits. Let ϕ:Λ/G→Λ/G denotethe homeomorphism induced by Φ on the orbit space. Subject tocertain conditions, we show that the set of topologically transitiveHölder(or Ck) equivariant homeomorphisms of Λ covering ϕ isopen and dense in Hölder (Ck) topology. Our results apply to skew andprincipal extensions by a compact connected semisimple Lie group over ageneral basic hyperbolic set.