Symmetric periodic Reeb orbits on the sphere

2022/11/29 by Miguel Abreu, Hui Liu, Abreu, Miguel +3
Mathematics · Physics and Astronomy · #37J10 #37J55 #53D40 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2211.16470

openalex publication_date 2022/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere S²ⁿ⁺¹ has at least n+1 simple periodic Reeb orbits. In this work, we consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least n+1 simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex.

Related

• Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere
• Closed Reeb orbits on the sphere and symplectically degenerate maxima
• Dynamical convexity and closed orbits on symmetric spheres
• Multiplicity of periodic orbits for dynamically convex contact forms
• Invariant Sets and Hyperbolic Closed Reeb Orbits