Center foliation: absolute continuity, disintegration and rigidity

TL;DR

This paper investigates the properties of center foliations in partially hyperbolic diffeomorphisms, revealing complex disintegration behaviors and conditions under which smooth conjugacy to linear models can be achieved.

Contribution

It provides new insights into the absolute continuity and disintegration of volume on center leaves, and establishes conditions for smooth conjugacy beyond absolute continuity.

Findings

Disintegration of volume on center leaves can be neither atomic nor Lebesgue.

Absolute continuity does not guarantee smooth conjugacy with linear models.

Stronger conditions on the center foliation lead to smooth conjugacy.

Abstract

In this paper we address the issues of absolute continuity for the center foliation (as well as the disintegration on the non-absolute continuous case) and rigidity of volume preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov on $\mathbb T^3$. It is shown that the disintegration of volume on center leaves may be neither atomic nor Lebesgue. It is also obtained results concerning the atomic disintegration. Moreover, the absolute continuity of the center foliation does not imply smooth conjugacy with its linearization. Imposing stronger conditions besides absolute continuity on the center foliation, smooth conjugacy is obtained.