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The sharp $C^0$-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces

Baptiste Serraille

TL;DR

The paper addresses the sharp $C^0$-fragmentation problem for Hamiltonian diffeomorphisms and for the kernel of the mass-flow homomorphism on closed surfaces. It introduces a triangulation-based fragmentation framework that yields a Lipschitz (i.e., fully linear) bound on the $C^0$-norms of the fragments relative to the original map, improving upon previous Hölder-type bounds. Central to the method are area-preserving extension lemmas on annuli (and rectangles) and two obstructions, $\\mathcal{O}$ and $\\mathcal{A}$, which are controlled via a refined Moser-type adjustment, enabling stepwise fragmentation along the skeleton of a triangulation. The results produce $C^0$-controlled isotopies in $\\overline{\\mathrm{Ham}}(\\Sigma,\\mu)$ (and in $\\mathrm{Ker}(\\theta)$ for the measure-preserving case) and resolve questions posed by Buhovsky and Seyfaddini, with implications for the structure and continuity properties of area-preserving dynamics on surfaces.

Abstract

In this paper, we present a $C^0$-fragmentation property for Hamiltonian diffeomorphisms. More precisely, it is known that for a given open covering $\mathcal{U}$ of a compact symplectic surface we can write each $C^0$-small enough Hamiltonian diffeomorphism as the composition of Hamiltonian diffeomorphisms compactly supported inside the open sets of the covering $\mathcal{U}$. We show that such a decomposition can be done with a Lipschitz estimate on the $C^0$-norm of the fragments. We also show the same property for the kernel of $θ$, the mass-flow homomorphism for homeomorphisms. This answers a question from Buhovsky and Seyfaddini.

The sharp $C^0$-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces

TL;DR

The paper addresses the sharp -fragmentation problem for Hamiltonian diffeomorphisms and for the kernel of the mass-flow homomorphism on closed surfaces. It introduces a triangulation-based fragmentation framework that yields a Lipschitz (i.e., fully linear) bound on the -norms of the fragments relative to the original map, improving upon previous Hölder-type bounds. Central to the method are area-preserving extension lemmas on annuli (and rectangles) and two obstructions, and , which are controlled via a refined Moser-type adjustment, enabling stepwise fragmentation along the skeleton of a triangulation. The results produce -controlled isotopies in (and in for the measure-preserving case) and resolve questions posed by Buhovsky and Seyfaddini, with implications for the structure and continuity properties of area-preserving dynamics on surfaces.

Abstract

In this paper, we present a -fragmentation property for Hamiltonian diffeomorphisms. More precisely, it is known that for a given open covering of a compact symplectic surface we can write each -small enough Hamiltonian diffeomorphism as the composition of Hamiltonian diffeomorphisms compactly supported inside the open sets of the covering . We show that such a decomposition can be done with a Lipschitz estimate on the -norm of the fragments. We also show the same property for the kernel of , the mass-flow homomorphism for homeomorphisms. This answers a question from Buhovsky and Seyfaddini.
Paper Structure (18 sections, 25 theorems, 74 equations, 4 figures)

This paper contains 18 sections, 25 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. The $C^0$-fragmentation property on surfaces
  3. The $C^0$-fragmentation for the group of Hamiltonian diffeomorphisms
  4. The $C^0$-fragmentation for the kernel of the mass-flow homomorphism
  5. Organization of the paper
  6. Acknowledgments
  7. Some definitions and notations
  8. Proof of some consequences of the $C^0$-fragmentation property
  9. Proof of the $C^0$-fragmentation property
  10. Covering associated to a triangulation
  11. Two corollaries of the area-preserving extension lemma for the annulus
  12. Definition of two obstructions
  13. Proof of Theorems \ref{['thm: fragmentation lemma on surfaces with Lipschitz estimate']} and \ref{['thm: fragmentation lemma for Homeo']}
  14. Proof of the area-preserving extension lemma
  15. Preliminaries
  16. ...and 3 more sections

Key Result

Corollary 1

Let $(\Sigma, \omega)$ be a closed symplectic surface and $d$ the distance induced by some Riemannian metric on $\Sigma$. There exists a constant $C>0$, such that for all $\phi$ in the group of Hamiltonian diffeomorphisms, there exists a Hamiltonian isotopy $\{\phi_t\}$ such that $\phi_0=Id$, $\phi_

Figures (4)

  • Figure 1: The triangulation $T$ is represented in black, the boundary of the open sets of the form $V_i^0$ are painted in red, the boundary of the open sets of the form $V_i^1$ are painted in green and the boundary of the open sets of the form $V_i^2$ are painted in blue.
  • Figure 2: The obstruction $\mathcal{O}$ is the signed area between the red curve and the blue curve.
  • Figure 3: Illustration of the proof of the extension lemma.
  • Figure 4: We have represented two fundamental domains. In red we drew the boundary of the image of a fundamental domain under $\varphi$, the cylinders $\mathcal{C}_1$ and $\mathcal{C}_3$ are colored in pink

Theorems & Definitions (55)

  • Corollary 1
  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Lemma 1.3: Area-preserving extension lemma for the annulus
  • Remark 1.4
  • Theorem 2
  • Remark 1.5
  • Corollary 2
  • Lemma 1.6
  • ...and 45 more