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arXiv · 1110.3782

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

Umberto Hryniewicz·Al Momin·Pedro A. S. Salomão
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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 $S^2$.

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Umberto Hryniewicz, Al Momin, Pedro A. S. Salomão. 2014-04-02. A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$. https://doi.org/10.1007/s00222-014-0515-2

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