Closed Orbits of Dynamically Convex Reeb Flows: Towards the HZ- and Multiplicity Conjectures

TL;DR

This paper proves new lower bounds on the number of prime closed Reeb orbits on star-shaped domains, advancing understanding of the multiplicity problem and related conjectures in symplectic dynamics.

Contribution

It establishes that such flows have at least n prime closed orbits, and in symmetric, non-degenerate cases, exactly n or infinitely many, resolving longstanding open questions.

Findings

Proved at least n prime closed Reeb orbits exist.

In symmetric, non-degenerate cases, flows have either n or infinitely many orbits.

Settled a case of the contact Hofer-Zehnder conjecture.

Abstract

We study the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in $\mathbb{R}^{2n}$. The first of our two main results asserts that such a flow has at least $n$ prime closed Reeb orbits, improving the previously known lower bound by a factor of two and settling a long-standing open question. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly $n$ or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks' celebrated $2$-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture. The proofs are based on several auxiliary results of independent interest on the structure of the filtered symplectic homology and the properties of closed orbits.