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Periodicity characterization by capacities for star-shaped domains

Jean Gutt, Vinicius G. B. Ramos, Shira Tanny

TL;DR

The paper advances the spectral classification of Reeb flows on boundaries of star-shaped domains by establishing that Besse flows correspond to blocks of equal equivariant symplectic-homology capacities, with the block length determined by the Morse–Bott data of the period orbit manifold. Using Morse–Bott indices, mean indices, and the $S^1$-equivariant SH barcode, the authors prove dynamic convexity for Besse domains and show that the SH spectrum aligns with the periods of the flow; this yields concrete criteria: a common equality of the first $n$ capacities implies Zoll, while a single orbit period yields a structured capacity chain of length $n$ and a precise index relation $k= frac12(\mu_-(S_\mathcal{T})-n+1)$. The work also identifies the Morse–Bott fixed-point sets $Y_T$ as ${\mathbb{F}}_2$-homology spheres and proves a diagonal SH barcode, enabling a complete spectral characterization in terms of capacities. Collectively, the results extend Ginzburg–Gürel–Mazzucchelli’s framework to general star-shaped domains (beyond convex cases) and provide a robust toolkit for understanding Zoll/Besse phenomena via symplectic-homology invariants.

Abstract

We complete the spectral characterization of Besse and Zoll Reeb flows on the standard contact sphere $S^{2n-1}$ initiated by Ginzburg-Gürel-Mazzucchelli. Roughly speaking, it states that a Reeb flow on the boundary of any star-shaped domain in $\mathbb{R}^{2n}$ is Besse if and only if it has $n$ coinciding Ekeland-Hofer capacities, and that it is Zoll if and only if the first $n$ capacities coincide.

Periodicity characterization by capacities for star-shaped domains

TL;DR

The paper advances the spectral classification of Reeb flows on boundaries of star-shaped domains by establishing that Besse flows correspond to blocks of equal equivariant symplectic-homology capacities, with the block length determined by the Morse–Bott data of the period orbit manifold. Using Morse–Bott indices, mean indices, and the -equivariant SH barcode, the authors prove dynamic convexity for Besse domains and show that the SH spectrum aligns with the periods of the flow; this yields concrete criteria: a common equality of the first capacities implies Zoll, while a single orbit period yields a structured capacity chain of length and a precise index relation . The work also identifies the Morse–Bott fixed-point sets as -homology spheres and proves a diagonal SH barcode, enabling a complete spectral characterization in terms of capacities. Collectively, the results extend Ginzburg–Gürel–Mazzucchelli’s framework to general star-shaped domains (beyond convex cases) and provide a robust toolkit for understanding Zoll/Besse phenomena via symplectic-homology invariants.

Abstract

We complete the spectral characterization of Besse and Zoll Reeb flows on the standard contact sphere initiated by Ginzburg-Gürel-Mazzucchelli. Roughly speaking, it states that a Reeb flow on the boundary of any star-shaped domain in is Besse if and only if it has coinciding Ekeland-Hofer capacities, and that it is Zoll if and only if the first capacities coincide.

Paper Structure

This paper contains 12 sections, 23 theorems, 94 equations.

Table of Contents

  1. Introduction and results
  2. Preliminaries
  3. The Conley--Zehnder index in the Morse--Bott setting
  4. Morse--Bott star-shaped domains
  5. Indices of paths of symplectic matrices
  6. Indices of closed orbits and Morse--Bott families.
  7. Symplectic homology
  8. The equivariant SH capacities
  9. Besse domains are dynamically convex
  10. Symplectic Homology of Besse contact forms
  11. Proofs of the main Theorems
  12. SH of a Morse--Bott family

Key Result

Theorem 1

If $W$ is star-shaped with discrete action spectrum and $c_{k}^{EH} (W) = \cdots = c_{k+n-1}^{EH}(W)$ for some $k\in {\mathbb{N}}$ then the Reeb flow on $\partial W$ is Besse and $c_k^{EH}(W)$ is a common period of the flow.

Theorems & Definitions (44)

  • Theorem : ginzburg2021spectral
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 2.1: Finiteness
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • ...and 34 more