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Hamiltonian Braids via Generating Functions

Francesco Morabito

Abstract

Given a compactly supported Hamiltonian diffeomorphism of the plane, one can define a generating function for it. In this paper, we show how generating functions retain information about the braid type of collections of fixed points of Hamiltonian diffeomorphisms. One the one hand, we show that it is possible to define a filtration keeping track of linking numbers of pairs of fixed points on the Morse complex of the generating function. On the other, we provide a finite-dimensional proof of a Theorem by Alves and Meiwes about the lower-semicontinuity of the topological entropy with respect to the Hofer norm. The technical tools come from work by Le Calvez which was developed in the 90s. In particular, we apply a version of positivity of intersections for generating functions.

Hamiltonian Braids via Generating Functions

Abstract

Given a compactly supported Hamiltonian diffeomorphism of the plane, one can define a generating function for it. In this paper, we show how generating functions retain information about the braid type of collections of fixed points of Hamiltonian diffeomorphisms. One the one hand, we show that it is possible to define a filtration keeping track of linking numbers of pairs of fixed points on the Morse complex of the generating function. On the other, we provide a finite-dimensional proof of a Theorem by Alves and Meiwes about the lower-semicontinuity of the topological entropy with respect to the Hofer norm. The technical tools come from work by Le Calvez which was developed in the 90s. In particular, we apply a version of positivity of intersections for generating functions.
Paper Structure (26 sections, 27 theorems, 195 equations, 1 figure)

This paper contains 26 sections, 27 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Relations to other work
  4. Acknowledgments
  5. Braid groups
  6. Twist Maps and their Generating Functions
  7. Generating functions quadratic at infinity
  8. Basic definitions and properties
  9. Morse theory for GFQIs on $\mathbb{R}^{2n}$
  10. Continuation maps
  11. Setup: twist generating functions and linking
  12. Construction of the generating function
  13. Collections of points in $E$ as braids
  14. Le Calvez's $h$ is a GFQI
  15. Palais-Smale condition
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $\varphi$ be a compactly supported Hamiltonian diffeomorphism of the plane with its standard symplectic form $dx\wedge dy$. Assume $\varphi$ is non degenerate on the interior of its support. Then for any generating function quadratic at infinity $S: \mathbb{R}^2\times \mathbb{R}^k\rightarrow \ma into an increasing filtration of the tensor complex. Here, $x$ and $y$ are both critical points of

Figures (1)

  • Figure :

Theorems & Definitions (80)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 70 more