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

Erman Cineli, Viktor L. Ginzburg, Basak Z. Gurel

TL;DR

This work proves two central results for dynamically convex Reeb flows on boundaries of star-shaped domains in ${\mathbb R}^{2n}$: (i) any such flow has at least $n$ prime closed Reeb orbits, extending previous lower bounds; (ii) if the domain is centrally symmetric and the flow is non-degenerate, the flow has exactly $n$ or infinitely many prime closed orbits, establishing a higher-dimensional contact Hofer-Zehnder-type dichotomy. The authors develop a robust persistence-module framework for (equivariant) filtered and local symplectic homology, prove a key one-dimensionality result for the persistence module under finiteness of prime orbits, and introduce a refined index recurrence theorem that governs the behavior of Conley–Zehnder indices under iteration. Together these tools yield precise control over orbit visibility, action/mean-index ratios, and the structure of the barcode, enabling a concrete ellipsoid comparison and a proof of a contact-analogue of Franks’ theorem. The results advance the understanding of Reeb pseudo-rotations and provide a rigorous bridge between symplectic topology and dynamical consequences for high-dimensional contact flows.

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.

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

TL;DR

This work proves two central results for dynamically convex Reeb flows on boundaries of star-shaped domains in : (i) any such flow has at least prime closed Reeb orbits, extending previous lower bounds; (ii) if the domain is centrally symmetric and the flow is non-degenerate, the flow has exactly or infinitely many prime closed orbits, establishing a higher-dimensional contact Hofer-Zehnder-type dichotomy. The authors develop a robust persistence-module framework for (equivariant) filtered and local symplectic homology, prove a key one-dimensionality result for the persistence module under finiteness of prime orbits, and introduce a refined index recurrence theorem that governs the behavior of Conley–Zehnder indices under iteration. Together these tools yield precise control over orbit visibility, action/mean-index ratios, and the structure of the barcode, enabling a concrete ellipsoid comparison and a proof of a contact-analogue of Franks’ theorem. The results advance the understanding of Reeb pseudo-rotations and provide a rigorous bridge between symplectic topology and dynamical consequences for high-dimensional contact flows.

Abstract

We study the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in . The first of our two main results asserts that such a flow has at least 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 or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks' celebrated -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.

Paper Structure

This paper contains 36 sections, 27 theorems, 184 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Application: ellipsoid comparison
  5. Symplectic homology
  6. Conventions and notation
  7. Graded persistence modules
  8. Filtered symplectic homology
  9. Local symplectic homology
  10. Basic definitions and facts
  11. Behavior under iterations
  12. From local to filtered homology
  13. The beginning and end maps
  14. Conley-Zehnder index and index recurrence
  15. Background: the Conley--Zehnder index and dynamical convexity
  16. ...and 21 more sections

Key Result

Theorem 1

Assume that the Reeb flow on the boundary $M^{2n-1}\subset {\mathbb R}^{2n}$ of a star-shaped domain is dynamically convex and has finitely many prime closed orbits, i.e., the flow is a Reeb pseudo-rotation. As a consequence, the Reeb flow on the boundary of a star-shaped domain has at least $n$ prime closed orbits whenever the flow is dynamically convex.

Theorems & Definitions (69)

  • Conjecture : The $n$-or-$\infty$ conjecture
  • Remark 1.1: Smoothness
  • Theorem 1: Multiplicity
  • Theorem 2: HZ-conjecture
  • Remark 1.2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Remark 1.3
  • Corollary 6
  • ...and 59 more