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.