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

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

2011-10-17

arxiv.org/abs/1110.3782v1

Mathematics

Dynamical Systems

58 pages

Scientific paper

We consider Reeb flows on the tight 3-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers of 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 $S^2$.

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