Disintegration of Invariant Measures for Hyperbolic Skew Products

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 H\"older and secondly in the case where $v$ is $\mathcal{C}^{1}$.