COCYCLES OVER PARTIALLY HYPERBOLIC MAPS par

A diffeomorphism f : M → M on a compact manifold M is partially hyperbolic if there exists a continuous, nontrivial Df -invariant splitting TxM = E s x ⊕ E c x ⊕ E u x , x ∈ M of the tangent bundle such that the derivative is a contraction along E and an expansion along E, with uniform rates, and the behavior of Df along the center bundle E is in between its behaviors along E and E, again by a uniform factor. Partial hyperbolicity is a natural generalization of the notion of uniform hyperbolicity (Anosov or even Axiom A, see [26]), that includes many interesting additional examples, most notably: diffeomorphisms derived from Anosov through deformation by isotopy, many affine maps on homogeneous spaces, certain skew-products over hyperbolic maps, and time-1 maps of Anosov flows. Partial hyperbolicity is an open condition, so any C small perturbation of these examples is partially hyperbolic as well. The stable and unstable bundles, E and E, are uniquely integrable; that is, there exist unique f -invariant foliations W and W tangent to E and E, respectively, at