Status  Confirmed 
Seminar Series  MATHIHES 
Subjects  math 
Date  Tuesday 29 January 2019 
Time  14:30 
Institute  IHES 
Seminar Room  Amphithéâtre Léon Motchane 
Speaker's Last Name  Ulcigrai 
Speaker's First Name  Corinna 
Speaker's Email Address  
Speaker's Institution  University of Zurich & University of Bristol 
Title  Chaotic Properties of Area Preserving Flows (1/4) 
Abstract  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. 
arXiv Preprint Number  
Comments  Cours de l'IHES 
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