Statut  Confirmé 
Série  MATHIHES 
Domaines  math 
Date  Mardi 29 Janvier 2019 
Heure  14:30 
Institut  IHES 
Salle  Amphithéâtre Léon Motchane 
Nom de l'orateur  Ulcigrai 
Prenom de l'orateur  Corinna 
Addresse email de l'orateur  
Institution de l'orateur  University of Zurich & University of Bristol 
Titre  Chaotic Properties of Area Preserving Flows (1/4) 
Résumé  Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudoperiodic topology. In this course we will focus on smooth areapreserving or locally Hamiltonian flows and their ergodic properties. The course will be selfcontained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner's work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multidimensional continued fraction algorithm (RauzyVeech induction): we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties. 
Numéro de preprint arXiv  
Commentaires  Cours de l'IHES 
Fichiers attachés 
Pour obtenir l' affiche de ce séminaire : [ Postscript  PDF ]

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