#### Volume 25, issue 6 (2021)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Barcodes and area-preserving homeomorphisms

### Frédéric Le Roux, Sobhan Seyfaddini and Claude Viterbo

Geometry & Topology 25 (2021) 2713–2825
##### Abstract

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 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.

##### Keywords
barcode, area-preserving homeomorphisms, symplectic topology, topological dynamics
##### Mathematical Subject Classification 2010
Primary: 37E30, 53D05, 53D40
##### Publication
Received: 21 November 2018
Revised: 17 September 2020
Accepted: 16 October 2020
Published: 30 November 2021
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Paul Seidel
##### Authors
 Frédéric Le Roux Institut de Mathématiques de Jussieu – Paris Rive Gauche UMR7586 Sorbonne Université Paris France Sobhan Seyfaddini CNRS Institut de Mathématiques de Jussieu – Paris Rive Gauche UMR7586 Sorbonne Université Paris France Claude Viterbo DMA UMR8553 du CNRS Ecole Normale Supérieure – PSL University Paris France