Anosov flows and Liouville pairs in dimension three

TL;DR

This paper establishes a homotopy correspondence between three-dimensional Anosov flows and contact form pairs called Anosov Liouville pairs, offering new insights into their classification and related structures.

Contribution

It introduces the concept of Anosov Liouville pairs and extends known correspondences to include projectively Anosov flows and bi-contact structures.

Findings

Every Anosov flow induces a homotopy class of Liouville structures on R×M.

The work extends the classification framework for Anosov flows in dimension three.

Provides a new perspective on the topology of Anosov flows and contact structures.

Abstract

Building upon the work of Mitsumatsu and Hozoori, we establish a complete homotopy correspondence between three-dimensional Anosov flows and certain pairs of contact forms that we call Anosov Liouville pairs. We show a similar correspondence between projectively Anosov flows and bi-contact structures, extending the work of Mitsumatsu and Eliashberg-Thurston. As a consequence, every Anosov flow on a closed oriented three-manifold $M$ gives rise to a Liouville structure on $\mathbb{R} \times M$ which is well-defined up to homotopy, and which only depends on the homotopy class of the Anosov flow. Our results also provide a new perspective on the classification problem of Anosov flows in dimension three.