Smooth Models for Certain Fibered Partially Hyperbolic Systems

TL;DR

This paper demonstrates that certain fibered partially hyperbolic systems over nilmanifolds can be smoothly modeled as isometric on fibers and relate to hyperbolic automorphisms, extending known conjugacy results.

Contribution

It establishes a leaf conjugacy to smooth models for fibered partially hyperbolic systems over nilmanifolds, generalizing conjugacy results for Anosov homeomorphisms.

Findings

Fibered partially hyperbolic systems are leaf conjugate to smooth models.

Anosov homeomorphisms of nilmanifolds are topologically conjugate to hyperbolic automorphisms.

The results extend conjugacy theory in the context of nilmanifolds.

Abstract

We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.