A note on stable holonomy between centers

There is a slight disparity in smooth ergodic theory, between Pesin the-ory 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 C1+δ hypothesis, whereas the Pugh-Shub stable ergodicity theorem assumes a C2 hypothesis. The purpose of this paper is to close the gap between C1+δ and C2. 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 = Eu⊕Ec⊕Es, 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 γ ̂ with ν, ν ̂ < 1 and ν < γ ≤ γ̂−1 < ν̂−1 (1) such that, for a unit vector v ∈ TpM, ‖Tfv ‖ ≤ ν(p), if v ∈ Es(p), (2) γ(p) ≤ ‖Tfv ‖ ≤ γ̂(p)−1, if v ∈ Ec(p), (3) ν̂(p)−1 ≤ ‖Tfv‖, if v ∈ Eu(p). (4) A partially hyperbolic diffeomorphism is dynamically coherent if there are foliations Wcs and Wcu tangent to Ec ⊕ Es and Ec ⊕ Eu respectively. In this case there is also a foliation Wc tangent to Ec whose leaves are the intersections of the leaves of Wcs and Wcu. Each leaf of Wcs is foliated by leaves of Wc and Ws; leaves of Wcu have the analogous property. 1 Theorem 0.1 ([PSW], Theorem B) Let f: M → M be a C2, partially hyperbolic, dynamically coherent diffeomorphism of a compact manifold M satisfying ν < γγ̂. Then the stable distribution is a uniformly C1 bundle when restricted to any leaf of the center-stable foliation. Consequently, the stable holonomy map between center transversals is C1. If the assumption that f is C2 is relaxed, then the first conclusion of Theorem 0.1 does not hold; that is, the stable bundle may no longer be a C1 bundle over every leaf of the center-stable foliation. This is easy to see in the case where the unstable bundle is trivial. Let F: T2 → T2 be a linear Anosov diffeomorphism of the 2-torus, and let f = ϕ ◦ F ◦ ϕ−1, where ϕ: T2 → T2 is a C1+δ diffeomorphism that is not C2. Then f is an Anosov diffeomorphism, with Esf = ϕ∗(