Disintegration of Invariant Measures for Hyperbolic Skew Products

TL;DR

This paper investigates how SRB measures for hyperbolic skew products can be disintegrated into stable manifold-supported measures, preserving smoothness of observables under mild conditions.

Contribution

It establishes that the disintegration process preserves the smoothness of observables, such as Hölder and ^1 functions, in hyperbolic skew products.

Findings

Disintegration preserves smoothness of observables.

Results apply to Hölder and ^1 functions.

Provides a method to relate observables on the skew product to quotiented systems.

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