Partial hyperbolicity and foliations in $\mathbb{T}^3$

TL;DR

This paper investigates the stability of dynamical coherence in partially hyperbolic diffeomorphisms on the 3-torus, establishing conditions under which such systems are either coherent or contain invariant tori, with broader implications for foliation theory.

Contribution

It proves that dynamical coherence is an open and closed property in this setting and classifies strong partially hyperbolic diffeomorphisms based on invariant tori or coherence.

Findings

Dynamical coherence is stable under perturbations.

Strong partially hyperbolic diffeomorphisms are either coherent or have invariant tori.

Develops general results on codimension one foliations.

Abstract

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.