Le prochain séminaire Nantes-Orsay se tiendra à Nantes le vendredi 10 décembre 2021 au Laboratoire de Mathématiques Jean Leray en salle Eole. Il sera
centré sur les propriétés dynamiques des champs de Reeb en dimension trois (orbites périodiques, entropie, sections de Birkhoff).
10H00 - 10h30 : Café.
10h30 - 11h30 : Umberto Hryniewicz (Aachen University)
Title: Generic existence of Birkhoff sections for Reeb flows
Abstract: A foliation transverse to a flow can be a powerful tool to study its dynamics. In dimension three, the best scenario is when such a foliation is a (possibly rational) open book decomposition and the pages are Birkhoff sections. The effective study of such foliations was initiated by Hofer, Wysocki and Zehnder in the early 2000’s for non-degenerate Reeb flows on the tight 3-sphere, who established the existence of finite-energy foliations. Very recently Colin, Dehornoy and Rechtman obtained a powerful existence result of a special kind of foliation called broken book decomposition for non-degenerate Reeb flows on any closed 3-manifold, and derived several applications, including the statement that there are two or infinitely many periodic orbits. The goal of this talk is to explain how to use the broken books to obtain Birkhoff sections. The conditions we need for this procedure to work are C^\infty generic, and can be checked in several classes of explicit and relevant (possibly degenerate) Reeb flows. This is joint work with Vincent Colin, Pierre Dehornoy and Ana Rechtman.
14h00 - 15h30 : Vincent Colin (Nantes)
Title: Broken book decompositions and dynamics of Reeb vector fields in dimension 3
Abstract: In this talk we will explain the existence of broken book decompositions adapted to Reeb vector fields
and explain how to derive several dynamical consequences (genericity of topological entropy, a dichotomy of two or infinitely many periodic orbits, existence of Birkhoff sections for geodesic flows by Contreras-Mazzuchelli). This is joint work with Pierre Dehornoy and Ana Rechtman.
We will also discuss several open questions regarding degenerate flows, including a conjecture of Le Roux and Le Calvez. The talk will be run on an informal format and followed by discussions with the audience.
15h30-16h30 : Café (et discussions)
Les questions abordées sont susceptibles d'intéresser au-delà du cercle des topologues/symplecticiens et il ne faudra pas hésiter à nous rejoindre, notamment pour participer aux discussions de l'après-midi !