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. 
Viktor L. Ginzburg, Doris Hein, Umberto L. Hryniewicz, Leonardo   Macarini
math.SG math.DS