Lectures on Symmetric Attractors

3. Lecture III: Stable ergodicity of skew extensions Let f be a diieomorphism of the compact Riemannian manifold M. We say f is partially hyperbolic 9] if there is a continuous Tf-invariant splitting such that Tf expands E u , contracts E s and sup kT s p fk < inf m(T c p f); sup kT c p fk < inf m(T u p f): (Here m(A) = kA ?1 k ?1 .) If the center bundle E c is tangent to a C 1-foliation F of M, then partial hyperbolicity can be thought of as hyperbolicity transverse to the foliation F. Grayson, Pugh & Shub 27] have suggested that partial hyperbolicity provides a natural setting for stable ergodicity. For example, they proved that the time one-map of the ge-odesic ow on the unit tangent bundle of surface of constant negative curvature is stably ergodic within the class of volume preserving diieomorphisms. Thus ergodicity holds on an open subset of volume preserving diieomorphisms even though these diieomorphisms will typically not be structurally stable. More recently, Pugh & Shub 42] extended this result to general manifolds of constant negative curvature and Wilkinson 44] has proved stable ergodicity of the time one map for all negatively curved surfaces. All of these results are diicult to prove on account of the fact that leaves of the center foliation are typically non compact and so veriication of ergodicity under perturbation requires delicate estimates. 3.1. Skew extensions. Partially hyperbolic sets arise naturally in the study of diieomor-phisms equivariant with respect to a compact non-nite Lie group ?. The set of all ?-orbits determines a (singular) foliation G of M. If f : M!M is ?-equivariant, then G is f-invariant. Example 3.1. Suppose that f(?x) = ?x. It is easy to show that one can choose a ?-invariant Riemannian metric on M with respect to which Tf : T?x!T?x is an isometry. In this situation, it natural to require hyperbolicity transverse to the ?-orbit (normal hyperbol-icity) { see Figure 15. ~ More generally, if all ?-orbits have the same dimension { and so determine a non-singular foliation { it is natural to ask about hyperbolicity transverse to ?-orbits on all of M. We start by reviewing some recent results that apply when the action of ? is free and ? is Abelian. 3.2. Result of Adler-Kitchens-Shub. Let T n denote the n-dimensional torus (no group …