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