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Action and rotation number of periodic orbits of area-preserving annulus diffeomorphisms

Huadi Qu

TL;DR

The paper generalizes Hutchings’ action-rotation results from the disk to the annulus for area-preserving diffeomorphisms. It develops a framework linking the action function, Calabi invariant, and rotation vectors, and uses a disk-augmentation technique to transfer estimates to the annulus, yielding lower bounds on mean action and rotation for periodic orbits under boundary rotations and flux. The results provide existence guarantees for periodic orbits with specified action and rotation properties and illustrate the theory with explicit examples and conjectural extensions. The work situates these results within the broader context of Reeb dynamics, ECH, and twist-type phenomena for invariant measures.

Abstract

We study periodic orbits for area-preserving surface diffeomorphisms, particularly some global properities related to the action function and rotation numbers. We generalize recent works of Machel Hutchings [4], proving the existence of periodic orbits with certain action and rotation values.

Action and rotation number of periodic orbits of area-preserving annulus diffeomorphisms

TL;DR

The paper generalizes Hutchings’ action-rotation results from the disk to the annulus for area-preserving diffeomorphisms. It develops a framework linking the action function, Calabi invariant, and rotation vectors, and uses a disk-augmentation technique to transfer estimates to the annulus, yielding lower bounds on mean action and rotation for periodic orbits under boundary rotations and flux. The results provide existence guarantees for periodic orbits with specified action and rotation properties and illustrate the theory with explicit examples and conjectural extensions. The work situates these results within the broader context of Reeb dynamics, ECH, and twist-type phenomena for invariant measures.

Abstract

We study periodic orbits for area-preserving surface diffeomorphisms, particularly some global properities related to the action function and rotation numbers. We generalize recent works of Machel Hutchings [4], proving the existence of periodic orbits with certain action and rotation values.

Paper Structure

This paper contains 7 sections, 9 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Action function and rotation numbers
  3. Action function
  4. Rotation numbers
  5. Main result
  6. Examples
  7. Further problems

Key Result

Theorem 1.1

Let $\theta_{0}\in\mathbb{R}$, let $f$ be an area-preserving, orientation-preserving diffeomorphism of $D$ which agrees with rotation by angle $\theta_0$ near the boundary. for the fixed $\beta$, fixed the value of the corresopiding action function on the boundary as $g(\partial D^{2})=\theta_{0}$, Then $\inf\{\mathcal{A}(\gamma)|\gamma\in\mathcal{P}(f)\}\leq Cal(f).$

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Example 2.3
  • Lemma 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 14 more