Hamiltonian Braids via Generating Functions

TL;DR

This paper explores how generating functions associated with Hamiltonian diffeomorphisms encode braid types of fixed points, introduces a filtration related to linking numbers, and proves a lower-semicontinuity result for topological entropy.

Contribution

It introduces a filtration capturing linking numbers in generating functions and provides a finite-dimensional proof of a lower-semicontinuity theorem for topological entropy.

Findings

Filtration tracks linking numbers of fixed points.

Generating functions encode braid types.

Proves lower-semicontinuity of topological entropy.

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.