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

Abstract

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 C^r-topology for all r > 0 (here r need not be an integer and C¹ is replaced by Lipschitz). Moreover, when $r\ge2$, we show that there is a C²-open and C^r-dense subset of C^r-extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) $\mathbb{R}^m$-extensions, thereby generalizing a result of Niţică and Pollicott, and on stable mixing for suspension flows.