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 C^s diffeomorphism, s >= 2. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the C^r topology for all r in(0,s] (except that C^1 is replaced by Lipschitz). In particular, when 2 <= r <= s, we show that there is a C^2 open and C^r dense subset of C^r extensions that are ergodic.

We obtain similar results on stable transitivity for (non-compact) R^m-extensions, thereby generalizing a result of Nitica and Pollicott, and on stable mixing for suspension flows.

For preprint, email: mikefield@gmail.com

Professor Mike Field
Department of Mathematics
Imperial College
London SW7 2AZ