Disintegration of Invariant Measures for Hyperbolic Skew Products

Abstract

We study hyperbolic skew products and the disintegration of the SRB measure into measures supported on local stable manifolds. Such a disintegration gives a method for passing from an observable $v$ on the skew product to an observable $\bar{v}$ on the system quotiented along stable manifolds. Under mild assumptions on the system we prove that the disintegration preserves the smoothness of $v$, firstly in the case where $v$ is Hölder and secondly in the case where $v$ is ${\mathcal{C}}^1$.

Updates + fixed typos. Final accepted version. To appear in Israel J. Math