Measure-theoretical properties of center foliations

TL;DR

This paper investigates the measure-theoretical properties of center foliations in partially hyperbolic diffeomorphisms, revealing complex behaviors such as singular and atomic disintegrations, and providing new examples that challenge existing assumptions.

Contribution

It constructs stable examples of center foliations with singular but non-atomic disintegrations and shows cases where lower leafwise absolute continuity fails in mostly contracting center dynamics.

Findings

Disintegration of volume measure can be singular without being atomic.

Constructed examples where center foliation is not lower leafwise absolutely continuous.

Identified complex measure-theoretical behaviors in center foliations of partially hyperbolic systems.

Abstract

Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add to this theory by constructing stable examples for which the disintegration is singular without being atomic. In the context of diffeomorphisms with mostly contracting center direction, for which upper leafwise absolute continuity is known to hold, we provide examples where the center foliation is not lower leafwise absolutely continuous.