The cohomological equation for partially hyperbolic diffeomorphisms

TL;DR

This paper develops a theory for solving the cohomological equation in the context of accessible, partially hyperbolic diffeomorphisms, and proves a regularity result for submanifolds.

Contribution

It introduces a new framework for the existence and regularity of solutions to the cohomological equation in partially hyperbolic dynamics.

Findings

Established conditions for the existence of solutions.

Proved that certain homogeneous submanifolds are smoothly embedded.

Connected regularity of solutions to the cohomological equation with submanifold smoothness.

Abstract

We establish a theory for the existence and regularity of solutions to the cohomological equation over an accessible, partially hyperbolic diffeomorphism. As a by-product of our techniques, we show that for $r>1$, any $C^r$ homogeneous, locally compact submanifold of a $C^r$ manifold is in fact a $C^r$ submanifold.