Symmetric periodic Reeb orbits on the sphere

TL;DR

This paper investigates symmetric periodic Reeb orbits on the sphere, proving the existence of at least one such orbit generally, and at least two under the condition of dynamical convexity, advancing understanding of symmetric dynamics on contact spheres.

Contribution

It establishes the existence of symmetric periodic Reeb orbits on the sphere, including a lower bound of two orbits under dynamical convexity, refining the longstanding conjecture with symmetry considerations.

Findings

At least one symmetric periodic orbit exists for any contact form.

At least two symmetric closed orbits exist if the contact form is dynamically convex.

Progress towards the symmetric version of the contact sphere conjecture.

Abstract

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere $S^{2n+1}$ 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.