Smooth rigidity for 3-dimensional dissipative Anosov flows

TL;DR

This paper proves a rigidity result for 3D dissipative Anosov flows, showing that conjugacies are either smooth or relate SRB measures with opposite signs, with implications for local rigidity and classification of these flows.

Contribution

It establishes a dichotomy for conjugacies between certain 3D Anosov flows, extending rigidity results and analyzing the structure of conjugacy classes and SRB measures.

Findings

Conjugacies are either smooth or flip SRB measures.

Local rigidity holds on a dense subset of flows.

The Teichmüller space of conjugacy classes is stratified by regularity.

Abstract

We consider two transitive $3$-dimensional Anosov flows which do not preserve volume and which are continuously conjugate to each other. Then, disregarding certain exceptional cases, such as flows with $C^1$ regular stable or unstable distributions, we prove that either the conjugacy is smooth or it sends the positive SRB measure of the first flow to the negative SRB measure of the second flow and vice versa. We give a number of corollaries of this result. In particular, we establish local rigidity on a $C^1$-open $C^\infty$-dense subspace of transitive Anosov flows; we improve the classical de la Llave-Marco-Moriy\'on rigidity theorem for dissipative Anosov diffeomorphisms on the $2$-torus by merely assuming matching of (full) Jacobian data at periodic points; we also exhibit the first evidence that the Teichm\"uller space of smooth conjugacy classes of Anosov diffeomorphisms on the $2$-torus is well-stratified according to regularity.