Closed Reeb orbits on the sphere and symplectically degenerate maxima

2012/10/18 by Ginzburg, Viktor L., Hein, Doris, Hryniewicz, Umberto L. +1 · 1 citation
#37J10 #53D40 #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1210.5037

Abstract

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.

Cited by

• Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in ℝ²ⁿ

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