Umberto L. Hryniewicz

Local contact homology and applications

We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll-Meyers theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of non-hyperbolic periodic orbits.

Closed Reeb orbits on the sphere and symplectically degenerate maxima

We show that the existence of one simple closed Reeb orbit of a particular type (a symplectically degenerate maximum) forces the Reeb flow to have infinitely many periodic orbits. We use this result to give a different proof of a recent theorem of Cristofaro-Gardiner and Hutchings asserting that every Reeb flow on the standard contact three-sphere has at least two periodic orbits. Our methods are based on adapting the machinery originally developed for proving the Hamiltonian Conley conjecture to the contact setting.

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

Sharp systolic inequalities for Reeb flows on the three-sphere

S^2