Anosov flows in dimension 3: an outside look

TL;DR

This paper provides an outsider's overview of Anosov flows in 3D, emphasizing their connections with topology, geometry, symplectic, and contact structures, aiming to facilitate understanding of the subject's main results and questions.

Contribution

It offers a comprehensive outsider perspective on Anosov flows, highlighting their interactions with various geometric and topological aspects, including the Barbot-Fenley leaf space theory.

Findings

Overview of main results and questions in Anosov flows

Discussion of the Barbot-Fenley leaf space theory

Emphasis on connections with symplectic and contact geometry

Abstract

These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agust\'in Moreno. I took the invitation to present the subject as asking from an outsider view of the subject, given the fact that my research uses both ideas and results from the theory of Anosov flows. The point of view of the course is to provide an overview of the main results and questions in the subject, with emphasis on the interaction with topology, geometry, specially symplectic geometry and contact aspects of the theory. Some detail is given in the presentation of the Barbot-Fenley theory of leaf spaces. Hopefully the notes will contribute in gaining a working knowledge of the theory and its many beautiful connections.