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Dynamical extensions of Zoll to nonsmooth convex bodies

Pazit Haim-Kislev

TL;DR

This work extends the symplectic Zoll property to non-smooth convex bodies by analyzing the Ekeland–Hofer–Zehnder capacity under hyperplane cuts and by classifying non-smooth action-minimizing characteristics. It introduces cuts additivity as a dynamical extension and proves its equivalence to a topological Zoll notion under suitable hypotheses, while also establishing a robust H^1-compactness framework for action-minimizers modulo coisotropic-face motions. The paper develops a detailed taxonomy of non-smooth dynamics—extreme-ray motion, coisotropic-face motion, and isotropic gliding—and shows isotropic gliding does not occur in key polyhedral settings, enabling compactness results and a clear relation between local dynamical behavior and global capacity additivity. Through explicit examples (e.g., 24-cell, rotated cross-polytope, and simplex-cut domains), it demonstrates both generalized Zoll and non-additive phenomena, clarifying when Zoll-type extensions hold in non-smooth contexts. These results provide new tools for understanding maximizers of the systolic ratio in non-smooth convex geometry and connect dynamical and topological extensions of Zoll with broader symplectic capacity questions.

Abstract

The symplectic Zoll property of smooth convex domains in the classical phase space has been extensively studied in recent years and, in particular, has been shown to detect local maximizers of the systolic ratio. We propose a dynamical extension of this property to the non-smooth setting, related to the behavior of the Ekeland-Hofer-Zehnder capacity with respect to hyperplane cuts. Under certain hypotheses, we establish its equivalence to a known topological extension of the Zoll property. In this context, we study the space of non-smooth action-minimizing closed characteristics and show that their dynamical behavior is not as irregular as one might first expect. We classify several types of dynamical behaviors and derive a certain $H^1$-compactness result, which is of independent interest.

Dynamical extensions of Zoll to nonsmooth convex bodies

TL;DR

This work extends the symplectic Zoll property to non-smooth convex bodies by analyzing the Ekeland–Hofer–Zehnder capacity under hyperplane cuts and by classifying non-smooth action-minimizing characteristics. It introduces cuts additivity as a dynamical extension and proves its equivalence to a topological Zoll notion under suitable hypotheses, while also establishing a robust H^1-compactness framework for action-minimizers modulo coisotropic-face motions. The paper develops a detailed taxonomy of non-smooth dynamics—extreme-ray motion, coisotropic-face motion, and isotropic gliding—and shows isotropic gliding does not occur in key polyhedral settings, enabling compactness results and a clear relation between local dynamical behavior and global capacity additivity. Through explicit examples (e.g., 24-cell, rotated cross-polytope, and simplex-cut domains), it demonstrates both generalized Zoll and non-additive phenomena, clarifying when Zoll-type extensions hold in non-smooth contexts. These results provide new tools for understanding maximizers of the systolic ratio in non-smooth convex geometry and connect dynamical and topological extensions of Zoll with broader symplectic capacity questions.

Abstract

The symplectic Zoll property of smooth convex domains in the classical phase space has been extensively studied in recent years and, in particular, has been shown to detect local maximizers of the systolic ratio. We propose a dynamical extension of this property to the non-smooth setting, related to the behavior of the Ekeland-Hofer-Zehnder capacity with respect to hyperplane cuts. Under certain hypotheses, we establish its equivalence to a known topological extension of the Zoll property. In this context, we study the space of non-smooth action-minimizing closed characteristics and show that their dynamical behavior is not as irregular as one might first expect. We classify several types of dynamical behaviors and derive a certain -compactness result, which is of independent interest.

Paper Structure

This paper contains 8 sections, 19 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction and results
  2. Dynamical and topological extensions of the Zoll property
  3. Non-smooth characteristics
  4. Examples
  5. Further discussion on the Zoll property
  6. Non-smooth action minimizers
  7. Dynamical characterization of Cuts Additivity
  8. Cuts Additivity and Zoll

Key Result

Theorem A

Let $K \subset {\mathbb{R}}^{2n}$ be a generalized Zoll convex domain for which the characteristic dynamics is well defined. Then $K$ is cuts additive.

Figures (2)

  • Figure 1: An illustration of a hyperplane cut which cuts an action-minimizing closed characteristic $\gamma$ into action minimizers $\gamma_1,\gamma_2$ for the respective parts.
  • Figure 2: The blue and orange lines describe $c_{_{\rm EHZ}}(K_2), c_{_{\rm EHZ}}(K_1)$ respectively, while the yellow line is their sum. The purple line is the capacity of the pentagon product $c_{_{\rm EHZ}}(P \times T)$.

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem A: Generalized Zoll implies Cuts Additive
  • Definition 1.2
  • Corollary 1.3
  • proof
  • Proposition 1.4
  • Theorem B: Cuts Additive implies Generalized Zoll
  • Remark 1.5
  • Theorem C: Classification of systoles dynamics
  • Proposition 1.6
  • ...and 55 more