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

Ergodic Theory Dyn. Syst. 25 (2005) 517-551.

Michael Field, Ian Melbourne and Andrew Török

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 (except that C¹ is replaced by Lipschitz). Moreover, when r is at least 2, 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) R^m-extensions, thereby generalizing a result of Nitica & Pollicott, and on stable mixing for suspension flows.