Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line

TL;DR

This paper establishes a spectral rigidity theorem for $ ext{R}$-covered Anosov flows on 3-manifolds, linking orbit equivalences to fundamental group elements, and explores their contact geometric implications.

Contribution

It introduces a rigidity result for hyperbolic-like group actions on the line and applies it to classify orbit equivalences of $ ext{R}$-covered Anosov flows, connecting dynamics and contact geometry.

Findings

Characterization of orbit equivalences via fundamental group elements

Finiteness of contact Anosov flows on any manifold

Existence of manifolds with multiple distinct Anosov contact structures

Abstract

We prove a rigidity result for group actions on the line whose elements have what we call "hyperbolic-like" dynamics. Using this, we give a spectral rigidity theorem for $\mathbb{R}$-covered Anosov flows on 3-manifolds, characterizing orbit equivalent flows in terms of the elements of the fundamental group represented by periodic orbits. As consequences of this, we give an efficient criterion to determine the isotopy classes of self orbit equivalences of $\mathbb{R}$-covered Anosov flows, and prove finiteness of contact Anosov flows on any given manifold. In the appendix with Jonathan Bowden, we prove that orbit equivalences of contact Anosov flows correspond exactly to isomorphisms of the associated contact structures. This gives a powerful tool to translate results on Anosov flows to contact geometry and vice versa. We illustrate its use by giving two new results in contact geometry: the existence of manifolds with arbitrarily many distinct Anosov contact structures, answering a question of Foulon--Hasselblatt--Vaugon, and a virtual description of the group of contact transformations of a contact Anosov structure, generalizing a result of Giroux and Massot.