Pedro A. S. Salomão

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

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

3

SO(3)

A dynamical characterization of universally tight lens spaces

SO(3) and applies to geodesic flows of Finsler metrics on

Global properties of tight Reeb flows with applications to Finsler geodesic flows on S 2

S^2

Sharp systolic inequalities for Reeb flows on the three-sphere

S2.

A full family of multimodal maps on the circle

We consider Reeb dynamics on the 3-sphere associated to a tight contact form. Our main result gives necessary and sufficient conditions for a periodic Reeb orbit to bound a disk-like global section for the Reeb flow, when the contact form is assumed to be non-degenerate.