Multiplicity of periodic orbits for dynamically convex contact forms

2015/09/28 by Miguel Abreu, Abreu, Miguel, Leonardo Macarini +1 · 3 citations
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.DS #math.SG

paper · pdf · doi:10.48550/arxiv.1509.08441

Version 1: 25 pages. Version 2: minor corrections, 26 pages. Version 3: minor corrections, to appear in a special volume of the Journal of Fixed Point Theory and Applications in honour of Paul Rabinowitz

arxiv created 2016/11/02 · arxiv updated 2016/11/03

Abstract

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on Sⁿ carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.

Cited by

• Topological Obstructions to Dynamical Convexity
• On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in ℝ²ⁿ
• Closed Reeb orbits on contact type hypersurfaces in T^*Sⁿ

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