A Sm\"org\aa sbord of (bi)contact structures, Reeb flows and pseudo-Anosov flows

TL;DR

This paper explores the relationship between different Reeb flows on the same contact structure, presenting initial results and posing open questions in the context of symplectic geometry and Anosov flows.

Contribution

It introduces a new perspective on comparing Reeb flows within contact structures, highlighting open problems and conjectures in the field.

Findings

Initial results on Reeb flow relationships

Open questions about contact structure dynamics

Conjectures linking contact topology and flow dynamics

Abstract

This note was written for the proceedings of the conference "Symplectic Geometry and Anosov Flows" held in Heideleberg in July 2024. It is meant as an invitation to the study of certain families of contact structures, centering around the following question: ``How much can one relate the dynamics of two distinct Reeb flows of the same contact structure?'' We gather some results as well as state many more questions and conjectures around that theme.