# A Poincaré–Birkhoff theorem for tight Reeb flows on \(S^3\)

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## 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 \(S^2\).

## Notes

### Acknowledgments

This paper was partially developed while the authors visited the Institute for Advanced Study for the thematic year on Symplectic Dynamics. The authors would like to thank the IAS for the hospitality. This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. U.H. was partially supported by CNPq grant 309983/2012-6; A.M. was partially supported through a postdoc at the Purdue University Department of Mathematics. P.S. was partially supported by CNPq grant 303651/2010-5 and by FAPESP grant 2011/16265-8.

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