On mixing properties of compact group extensions of hyperbolic systems

Abstract

Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds.

On differentiability of SRB states for partially hyperbolic systems

“…By [20] a generic extension is rapidly mixing. (Corollary 6.5 of [20] gives (5) for S coming from Markov partition and [22], Proposition 4 extends it to arbitrary regular S. See also Appendix A of the present paper. )…”

“…Let Γ t (l 1 , l 2 ) denote the set {g(W )} for all chains W such that the number of legs n(W ) ≤ l 1 and for all j d W * (x j+1 , x j ) ≤ l 2 . Then ( [20], Section 4) f is rapidly mixing if and only if Γ t (l 1 , l 2 ) is Diophantine for large (l 1 , l 2 ), that is there are constants D, σ such that for each…”

“…Recall also that by (20) for all k, V ∈ V Cα k (M), and for all k, l, a l ∈ Cα k (M). These observations allow us to use Proposition 2 to estimate correlation functions containingρ, V and a l .…”

Spectrum and Statistical Properties of Chaotic Dynamics

“…The SRB measure is available for free, and in the volume preserving case mixing follows from work of Brin. Dolgopyat [45] combined key results of Brin and Katok on partially hyperbolic skew products with information gained from his previous study of hyperbolic flows [41,42,43] to prove that a diophantine condition on the Brin group implies rapid (i.e., faster than any polynomial) decay of correlations for C ∞ observables. The speed of mixing is not necessarily exponential for Hölder observables.…”

“…Other systems admitting an SRB measure with rapid mixing properties but not a single closed orbit are compact group extensions of hyperbolic systems as studied by Dolgopyat [45]. (Each of these discrete-time dynamical system without closed orbits has a single zero Lyapunov exponent and is imbedded in a hyperbolic flow with the "right" periodic orbit structure, the zeta function of which has the expected analytic properties [41]- [43].…”