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 (2)

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

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• Homological invariants of codimension 2 contact submanifolds