Atomic disintegrations for partially hyperbolic diffeomorphisms

TL;DR

This paper proves that for certain volume-preserving, partially hyperbolic diffeomorphisms on the 3-torus, the volume disintegration along central leaves is a single delta measure, extending previous atomic disintegration results.

Contribution

It establishes that in a class of volume-preserving partially hyperbolic diffeomorphisms, the disintegration along central leaves is a delta measure, generalizing earlier findings on atomic disintegrations.

Findings

Disintegration of volume along central leaves is a delta measure.

Provides conditions for delta measure disintegrations in circle bundle skew products.

Extends atomic disintegration results to a broader class of diffeomorphisms.

Abstract

Shub & Wilkinson and Ruelle & Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of circles. The foliation is not absolutely continuous, in fact, Ruelle & Wilkinson established that the disintegration of volume along central leaves is atomic. We show that in such a class of volume preserving diffeomorphisms the disintegration of volume along central leaves is a single delta measure. We also formulate a general result for conservative three dimensional skew product like diffeomorphisms on circle bundles, providing conditions for delta measures as disintegrations of the smooth invariant measure.