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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 C0 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
References
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