A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$

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 Poincaré-Birkhoff theorem on area-preserving annulus homeomorphisms. An analogous theorem holds on $SO(3)$ and applies to geodesic flows of Finsler metrics on $S^2$.

67 pages. To appear in Inventiones Mathematicae

References in corpus (3)

• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• Intersection theory of punctured pseudoholomorphic curves
• Contact Homology of Orbit Complements and Implied Existence

Cited by in corpus (6)

• A dynamical characterization of universally tight lens spaces
• Cylindrical contact homology and topological entropy
• A symplectic dynamics proof of the degree-genus formula
• Pseudorotations of the 2-disc and Reeb flows on the 3-sphere
• Global Hypersurfaces of Section for Geodesic Flows on Convex Hypersurfaces
• Homological invariants of codimension 2 contact submanifolds