Multiplicity of periodic orbits for dynamically convex contact forms

TL;DR

This paper establishes a sharp lower bound on the number of distinct contractible periodic orbits for certain Reeb flows, with implications for closed geodesics and periodic orbit multiplicities, using advanced index theory.

Contribution

It provides a new lower bound for periodic orbits in dynamically convex contact forms and extends results to non-aspherical cases, utilizing the common index jump theorem.

Findings

Lower bound for contractible periodic orbits in non-aspherical cases

At least two prime closed geodesics on bumpy Finsler spheres

Multiplicity results for elliptic and non-hyperbolic orbits

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 $S^n$ 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.