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

Abstract

<p>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 <i>C</i>² diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the <i>C^r</i>-topology for all <i>r</i> > 0 (here <i>r</i> need not be an integer and <i>C</i>¹ is replaced by Lipschitz). Moreover, when <i>r</i> is greater than or equal to 2, we show that there is a <i>C</i>²-open and <i>C^r</i>-dense subset of <i>C^r</i>-extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) <b>R</b>^<i>m</i>-extensions, thereby generalizing a result of Nitica and Pollicott, and on stable mixing for suspension flows.</p>

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