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About barcodes and Calabi invariant for Hamiltonian homeomorphisms of surfaces

Abstract : The goal of this thesis is to give some links between sympletic topology and the study of dynamical systems through the notion of barcodes of Hamiltonian homeomorphisms of surfaces and the Calabi invariant of Hamiltonian diffeomorphisms of the unit disk. These two objects represent powerful invariants in symplectic topology. More precisely, we aim at giving a dynamical interpretation of these objects. This thesis is divided into two parts. In a first part we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give some constructions of barcodes of some Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations. In particular, we will prove that for the simplest cases, our constructions coincide with the Floer Homology barcodes. In a second part we will deal with the Calabi invariant of the Hamiltonian diffeomorphisms of the unit disk. Inspired by the dynamical interpretation of this object developed by Fathi in his thesis, we will extend it to the group of C1 Hamiltonian diffeomorphisms of the disk. In particular, we will be able to compute the Calabi invariant of some irrational pseudo-rotations of the disk.
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Submitted on : Friday, December 3, 2021 - 5:27:07 PM
Last modification on : Tuesday, January 4, 2022 - 5:48:47 AM


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Benoît Joly. About barcodes and Calabi invariant for Hamiltonian homeomorphisms of surfaces. Dynamical Systems [math.DS]. Sorbonne Université, 2021. English. ⟨NNT : 2021SORUS196⟩. ⟨tel-03465434⟩



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