Dynamical convexity and closed orbits on symmetric spheres

TL;DR

This paper explores the dynamics of symmetric Reeb flows on spheres, introducing strong dynamical convexity, and proves existence of multiple closed orbits under certain conditions, with examples and relaxations of convexity assumptions.

Contribution

It introduces the notion of strong dynamical convexity for symmetric contact forms and establishes lower bounds on closed Reeb orbits, linking convexity to dynamical properties.

Findings

Any symmetric contact form with a weak convexity condition has at least n+1 simple closed Reeb orbits.

Strong dynamical convexity follows from convexity in antipodally symmetric contact forms.

Examples show that dynamical convexity does not imply strong dynamical convexity in higher dimensions.

Abstract

The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2n+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examples of antipodally symmetric dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has non-empty $C^1$-interior.