A note on stable holonomy between centers

There is a slight disparity in smooth ergodic theory, between Pesin theory and the Pugh-Shub partially hyperbolic theory. Both theories assume a weakened form of hyperbolicity and conclude ergodicity. While the two theories have considerable overlap, Pesin theory assumes a C hypothesis, whereas the Pugh-Shub stable ergodicity theorem assumes a C hypothesis. The purpose of this paper is to close the gap between C and C. Let f : M → M be a partially hyperbolic diffeomorphism of a compact manifold M . Partially hyperbolic means the following. There is a nontrivial splitting of the tangent bundle, TM = E⊕E⊕E, that is invariant under the derivative map Tf . Further, there is a Riemannian metric for which we can choose continuous positive functions ν, ν̂, and γ̂ with