Stable Ergodic Properties of Cocycles over Hyperbolic Attractors

  • C P Walkden
  • Published 1998

Abstract

We consider a hyperbolic ow t deened on an attracting basic set. A map from the rst (Cech) cohomology group of into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of t. x1 Introduction In PS] it is conjectured thatà little hyperbolicity goes a long way in guaranteeing stable ergodic behaviour'. By stably ergodic we mean the existence of an open dense set of suitable perturbations for which the perturbed dynamical system is ergodic. In particular, in PS] conditions are given for a volume preserving partially hyperbolic diieomorphism to be stably ergodic in the class of all volume preserving diieomorphisms. An example of a partially hyperbolic diieomorphism is given by a group extension or skew product of a hyperbolic diieomorphism. These are maps of the form f (x; y) = (x; f(x)y) on G where is a hyperbolic diieomorphism, G is a compact Lie group and f : ! G is smooth. One can ask how the ergodic properties of f depend on f. co-authors P2, PP1, FP]. When G is a torus, it is proved in PP1] that such skew products are stably ergodic and stably mixing. Analogous results when G is a compact connected Lie group are proved in FP]. In this paper we consider the case of continuous time. We prove: Theorem A Let t be a hyperbolic ow on a manifold with vector eld X and preserving a suitable measure. Let G be a compact connected Lie group and let F : R ! G be a map such that F t+s (x) = F t (s x)F s (x). Deene a ow on 1 G by t F (x; y) = (t x; F t (x)y). Then there is an open dense set of F (in an appropriate topology) for which t F is ergodic or mixing. The ow t F is the continuous time analogue of f. If t F is weak-mixing then, by a result in R], it is Bernoulli. For brevity, we say such a ow is mixing. We also prove the existence of an open dense set of mixing skew products. Observe that t F is a ow on G, a trivial G-bundle over , and that t F commutes with the (right) G-action on G. More generally, we could consider a manifold ^ …

Cite this paper

@inproceedings{Walkden1998StableEP, title={Stable Ergodic Properties of Cocycles over Hyperbolic Attractors}, author={C P Walkden}, year={1998} }