Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets

We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the Cr-topology for all r > 0 (here r need not be an integer and C1 is replaced by Lipschitz). Moreover, when r is greater than or equal to 2, we show that there is a C2-open and Cr-dense subset of Cr-extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) Rm-extensions, thereby generalizing a result of Nitica and Pollicott, and on stable mixing for suspension flows.