How typical are pathological foliations in partially hyperbolic dynamics: an example

TL;DR

This paper demonstrates that in a broad class of volume-preserving partially hyperbolic diffeomorphisms on the 3-torus, the central foliation is typically non-absolutely continuous, highlighting the typicality of pathological foliations.

Contribution

It provides the first example showing that non-absolutely continuous central foliations are generic in a large class of volume-preserving partially hyperbolic systems.

Findings

Central foliation is generically non-absolutely continuous in the studied class.

Non-compact central leaves are common in these systems.

The result highlights the typicality of pathological foliations in this setting.

Abstract

We show that for a large space of volume preserving partially hyperbolic diffeomorphisms of the 3-torus with non-compact central leaves the central foliation generically is non-absolutely continuous.