Barcodes and area-preserving homeomorphisms

TL;DR

This paper introduces the use of barcodes as a novel tool to analyze the dynamics of area-preserving homeomorphisms, establishing continuity properties and invariants related to fixed points and weak conjugacy.

Contribution

It develops a framework for applying barcode theory to Hamiltonian homeomorphisms, including defining barcodes for these maps and proving invariance properties under weak conjugacy.

Findings

Barcode of a Hamiltonian diffeomorphism depends continuously on the map.

Number of fixed points is a weak conjugacy invariant for certain Hamiltonian homeomorphisms.

Established a proof of the Isometry Theorem incorporating Barannikov's theory.

Abstract

In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.