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Low-dimensional topology and symplectic dynamics

Dan Cristofaro-Gardiner

TL;DR

The notes survey a program in low-dimensional symplectic dynamics that blends three-manifold invariants (PFH/ECH) with dynamical questions about Reeb and area-preserving flows. Central achievements include resolving the Simplicity Conjecture for compactly supported area-preserving diffeomorphisms via a finite Hofer energy subgroup and the Weyl-law recovery of Calabi invariants from PFH spectral invariants, as well as establishing Hofer–Wysocki–Zehnder’s two-or-infinity dichotomy under torsion assumptions. The approach hinges on PFH/ECH spectral invariants, their axioms, and a detailed analysis of low-action $J$-holomorphic curves and $U$-map towers to produce global surfaces of section and invariant-set structure, with Seiberg–Witten theory providing a complementary backbone in the Weyl-law regime. These methods yield consequences for geodesic and Reeb dynamics, inform entropy questions, and illuminate higher-dimensional prospects and open questions in the field.

Abstract

These are notes to accompany my lectures at the $2024$ "Current Developments in Mathematics" conference hosted by Harvard/MIT. The lectures were about some recent progress in our understanding of two and three dimensional dynamical systems, using in part some tools from low-dimensional topology. In these notes, I try to give a sense for how this works by discussing a few examples from problems I have worked on and surveying some related developments. Some symplectic variants of Weyl's law play a key role. I also briefly comment on some other developments and on the relationship with what is known in higher dimensions. The lectures themselves are available online.

Low-dimensional topology and symplectic dynamics

TL;DR

The notes survey a program in low-dimensional symplectic dynamics that blends three-manifold invariants (PFH/ECH) with dynamical questions about Reeb and area-preserving flows. Central achievements include resolving the Simplicity Conjecture for compactly supported area-preserving diffeomorphisms via a finite Hofer energy subgroup and the Weyl-law recovery of Calabi invariants from PFH spectral invariants, as well as establishing Hofer–Wysocki–Zehnder’s two-or-infinity dichotomy under torsion assumptions. The approach hinges on PFH/ECH spectral invariants, their axioms, and a detailed analysis of low-action -holomorphic curves and -map towers to produce global surfaces of section and invariant-set structure, with Seiberg–Witten theory providing a complementary backbone in the Weyl-law regime. These methods yield consequences for geodesic and Reeb dynamics, inform entropy questions, and illuminate higher-dimensional prospects and open questions in the field.

Abstract

These are notes to accompany my lectures at the "Current Developments in Mathematics" conference hosted by Harvard/MIT. The lectures were about some recent progress in our understanding of two and three dimensional dynamical systems, using in part some tools from low-dimensional topology. In these notes, I try to give a sense for how this works by discussing a few examples from problems I have worked on and surveying some related developments. Some symplectic variants of Weyl's law play a key role. I also briefly comment on some other developments and on the relationship with what is known in higher dimensions. The lectures themselves are available online.

Paper Structure

This paper contains 50 sections, 33 theorems, 94 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Terminology
  4. Two old problems
  5. Mysteries about the group of area-preserving homeomorphisms
  6. Two or infinity
  7. Density, periodic orbits, and other invariant sets
  8. Generic density of periodic orbits
  9. The Le Calvez - Yoccoz property
  10. The algebraic structure of homeomorphism groups
  11. Statement of results
  12. The diffeomorphism case
  13. Difficulties in extending Calabi
  14. A normal subgroup via Hofer geometry
  15. Recovering Calabi with the Weyl law
  16. ...and 35 more sections

Key Result

Theorem 1.6

irie A $C^{\infty}$-generic contact form on a closed three-manifold has the property that its Reeb orbits are dense.

Figures (7)

  • Figure 1: The Calabi invariant is not $C^0$-continuous.
  • Figure 2: The infinite twist
  • Figure 3: Approximating the infinite twist by smooth twists
  • Figure 4: The kind of curves counted by the PFH differential
  • Figure 5: An example of the corner rounding operation. In the combinatorial model, this represents a differential from an orbit set (in this case, consisting of two orbits) to another.
  • ...and 2 more figures

Theorems & Definitions (56)

  • Conjecture 1.2: "Weinstein conjecture"
  • Remark 1.3
  • Example 1.4
  • Conjecture 1.5: "Hofer-Wysocki-Zehnder's two or infinity conjecture"
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 2.1
  • Theorem 2.2
  • ...and 46 more