A Poincar\'e-Birkhoff theorem for tight Reeb flows on $S^3$

TL;DR

This paper proves a Poincaré-Birkhoff type theorem for Reeb flows on the tight 3-sphere, showing the existence of infinitely many periodic orbits under certain conditions, with applications to geodesic flows.

Contribution

It establishes a new linking and periodic orbit existence result for Reeb flows on the tight 3-sphere, extending classical Poincaré-Birkhoff ideas to contact dynamics.

Findings

Existence of infinitely many periodic trajectories with specific linking properties.

The result applies to Reeb flows with certain non-resonance conditions.

Analogous results are obtained for flows on SO(3) and geodesic flows on S^2.

Abstract

We consider Reeb flows on the tight $3$-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition then there exist infinitely many periodic trajectories distinguished by their linking numbers with the components of the link. This result admits a natural comparison to the Poincar\'e-Birkhoff theorem on area-preserving annulus homeomorphisms. An analogous theorem holds on $SO(3)$ and applies to geodesic flows of Finsler metrics on $S^2$.