Dynamical implications of convexity beyond dynamical convexity

TL;DR

This paper explores the dynamical consequences of convexity on symmetric spheres, revealing new types of periodic orbits and examples of contact forms that are not dynamically convex but share some properties with convex forms.

Contribution

It establishes sharp dynamical implications of convexity beyond dynamical convexity, including the existence of specific periodic orbits and new examples of contact forms with particular stability properties.

Findings

Existence of elliptic and non-hyperbolic periodic orbits.

Construction of new dynamically convex contact forms not contactomorphic to convex ones.

Demonstration of multiplicity of symmetric non-hyperbolic and other closed Reeb orbits.

Abstract

We establish sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to show the existence of elliptic and non-hyperbolic periodic orbits and to furnish new examples of dynamically convex contact forms, in any dimension, that are not equivalent to convex ones via contactomorphisms that preserve the symmetry. Moreover, these examples are $C^1$-stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $C^1$-close to those preserving the symmetry. We also show the multiplicity of symmetric non-hyperbolic and symmetric (not necessarily non-hyperbolic) closed Reeb orbits under suitable pinching conditions.