Barcodes for Hamiltonian homeomorphisms of surfaces

TL;DR

This paper introduces a new method to construct barcodes for Hamiltonian homeomorphisms of surfaces using graph theory, linking dynamical systems with Floer homology in a novel way.

Contribution

It presents a graph-based approach to define barcodes for Hamiltonian homeomorphisms, connecting maximal isotopies and Floer homology in surface dynamics.

Findings

Constructed barcodes from graphs associated to isotopies.

Proved equivalence with Floer homology barcodes in simple cases.

Provided a new dynamical perspective on Floer homology.

Abstract

In this article, the main goal is to give a dynamical point of view of Floer homology barcodes for Hamiltonian homeomorphisms of surfaces. More specifically, we describe a way to construct barcodes for Hamiltonian homeomorphisms of surfaces from graphs. We will define graphes associated to maximal isotopies of a Hamiltonian homeomorphism using Le Calvez's positively transverse foliation theory and to those graphs we will associate barcodes. In particular, we will prove that for the simplest cases, our constructions coincide with the Floer Homology barcodes.