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A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization

Lev Buhovsky, Ben Feuerstein, Leonid Polterovich, Egor Shelukhin

TL;DR

The paper addresses the growth of Hofer geometry under autonomous Hamiltonian flows on the sphere and proves an enhanced dichotomy: either the Hofer growth is linear or the flow remains uniformly bounded in Hofer distance. It introduces Hamiltonian symmetrization, a method to conjugate autonomous flows to height-function flows, and builds a symmetrization map $\Sigma$ that preserves growth rates via $\rho(H)=\rho(\Sigma(H))$ and yields a uniform bound when $\Sigma(H)=0$. The construction uses a two-pronged approach via flattened quasi-Morse functions and quasi-morphisms to obtain Lipschitz control and independence from the generating Hamiltonian, ultimately linking to a bounded-error conjugation in the completion of the Hofer group. The results illuminate the Hofer geometry of $\mathrm{Ham}(S^2)$, provide tools potentially extendable to other surfaces, and connect to Reeb-graph combinatorics and Floer-theoretic estimators.

Abstract

We prove that autonomous Hamiltonian flows on the two-sphere exhibit the following dichotomy: the Hofer norm either grows linearly or is bounded in time by a universal constant C. Our approach involves a new technique, Hamiltonian symmetrization. Essentially, we prove that every autonomous Hamiltonian diffeomorphism is conjugate to an element C-close in the Hofer metric to one generated by a function of the height.

A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization

TL;DR

The paper addresses the growth of Hofer geometry under autonomous Hamiltonian flows on the sphere and proves an enhanced dichotomy: either the Hofer growth is linear or the flow remains uniformly bounded in Hofer distance. It introduces Hamiltonian symmetrization, a method to conjugate autonomous flows to height-function flows, and builds a symmetrization map that preserves growth rates via and yields a uniform bound when . The construction uses a two-pronged approach via flattened quasi-Morse functions and quasi-morphisms to obtain Lipschitz control and independence from the generating Hamiltonian, ultimately linking to a bounded-error conjugation in the completion of the Hofer group. The results illuminate the Hofer geometry of , provide tools potentially extendable to other surfaces, and connect to Reeb-graph combinatorics and Floer-theoretic estimators.

Abstract

We prove that autonomous Hamiltonian flows on the two-sphere exhibit the following dichotomy: the Hofer norm either grows linearly or is bounded in time by a universal constant C. Our approach involves a new technique, Hamiltonian symmetrization. Essentially, we prove that every autonomous Hamiltonian diffeomorphism is conjugate to an element C-close in the Hofer metric to one generated by a function of the height.
Paper Structure (3 sections, 4 theorems, 8 equations, 1 figure)

This paper contains 3 sections, 4 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The symmetrization map
  3. Symmetrization of flat quasi-Morse functions

Key Result

Theorem 1.2

Consider the two-dimensional sphere $S^2$ equipped with a smooth area form. Then for every $F \in C^\infty(M)$ with $\rho(F)=0$ and all $t$ in ${\mathbb{R}}$ we have

Figures (1)

  • Figure :

Theorems & Definitions (8)

  • Theorem 1.2: Enhanced dichotomy
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5