Partially Hyperbolic Dynamics on $\mathbb T^4$: Existence of Compact Center-Unstable Leaves

TL;DR

This paper investigates the structure of partially hyperbolic diffeomorphisms on tori, establishing conditions for the existence of compact center-unstable leaves and their properties, with implications for the topology of these dynamical systems.

Contribution

It proves the existence of a transverse closed curve in the universal cover under certain conditions and shows that leaf conjugacy to the linear part implies no compact incompressible center-unstable submanifolds.

Findings

Existence of a transverse closed curve in the universal cover.

No compact incompressible center-unstable submanifold if leaf conjugate to linear part.

Incompressibility assumptions can be removed for $\

Abstract

We show that for $n\ge2$, if a partially hyperbolic diffeomorphism $f:\mathbb T^{n+1}\to \mathbb T^{n+1}$ with $\dim E^s=\dim E^c=1$ has an invariant center-unstable foliation with a compact incompressible leaf, then this foliation has a transverse closed curve in the universal cover. Also, if $f$ is leaf conjugate to its linear part, it has no compact incompressible center-unstable submanifold. In particular, by the incompressibility result we obtained on Anosov tori, the incompressibility assumptions can be removed when $f$ is defined on $\mathbb T^4$.