Smooth Models of Fibered Partially Hyperbolic Systems

TL;DR

This paper investigates conditions under which fibered partially hyperbolic systems can be smoothly fibered and explores obstructions to smooth lifts of Anosov diffeomorphisms, providing new examples and theoretical insights.

Contribution

It establishes criteria for homotoping fibered partially hyperbolic systems to smooth fiberings and identifies obstructions to smooth lifts of Anosov diffeomorphisms.

Findings

Homotopy to smooth fiberings under certain conditions

Existence of obstructions to smooth lifts of Anosov diffeomorphisms

Example of a bundle where lift is continuous but not smooth

Abstract

We study fibered partially hyperbolic diffeomorphisms. We show that as long as certain topological obstructions vanish and as long as homological minimum expansion dominates the distortion on the fibers that a fibered partially hyperbolic system can be homotoped to a fibered partially hyperbolic system with a $C^{\infty}$-center fibering. In addition, we study obstructions to the existence of smooth lifts of Anosov diffeomorphisms to bundles. In particular, we give an example of smooth topologically trivial bundle over a torus, where an Anosov diffeomorphism can lift continuously but not smoothly to the bundle.