Smooth rigidity for codimension one Anosov flows

TL;DR

This paper introduces a new technique using matching functions and periodic cycle functionals to improve the regularity of conjugacies in codimension one Anosov flows, showing that continuous conjugacies are often actually smooth.

Contribution

It develops the matching functions technique and demonstrates that continuous conjugacies between certain Anosov flows are generically smooth, advancing rigidity theory.

Findings

Simple periodic cycle functionals are $C^1$ regular for conservative codimension one Anosov flows.

Continuous conjugacies are $C^1$ for an open and dense set of flows.

The method improves understanding of smoothness properties of conjugacies in dynamical systems.

Abstract

We introduce the matching functions technique in the setting of Anosov flows. Then we observe that simple periodic cycle functionals (also known as temporal distance functions) provide a source of matching functions for conjugate Anosov flows. For conservative codimension one Anosov flows $\varphi^t\colon M\to M$, $\dim M\ge 4$, these simple periodic cycle functionals are $C^1$ regular and, hence, can be used to improve regularity of the conjugacy. Specifically, we prove that a continuous conjugacy must, in fact, be a $C^1$ diffeomorphism for an open and dense set of codimension one conservative Anosov flows.