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SUMMARY:Chaotic Properties of Area Preserving Flows (1/4)
DTSTART;VALUE=DATE-TIME:20190129T133000Z
DTEND;VALUE=DATE-TIME:20190129T153000Z
DTSTAMP;VALUE=DATE-TIME:20190919T231736Z
UID:indico-event-4274@indico.math.cnrs.fr
DESCRIPTION:\n\nFlows on surfaces are one of the fundamental examples of d
ynamical systems\, studied since Poincaré\; area preserving flows arise f
rom many physical and mathematical examples\, such as the Novikov model of
electrons in a metal\, unfolding of billiards in polygons\, pseudo-period
ic topology. In this course we will focus on smooth area-preserving -or lo
cally Hamiltonian- flows and their ergodic properties. The course will be
self-contained\, 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 i
nclude the full classification of generic mixing properties (mixing\, weak
mixing\, absence of mixing) motivated by a conjecture by Arnold\, up to v
ery recent rigidity and disjointness results\, which are based on a breakt
hrough adaptation of ideas originated from Marina Ratner's work on unipote
nt flows to the context of flows with singularities. We will in particular
highlight the role played by shearing as a key geometric mechanism whic
h explains many of the chaotic properties in this setup. A key tool is pro
vided by Diophantine conditions\, which\, in the context of higher genus s
urfaces\, are imposed through a multi-dimensional continued fraction algor
ithm (Rauzy-Veech induction): we will explain how and why they appear an
d how they allow to prove quantitative shearing estimates needed to invest
igate chaotic properties.\n\n\n\nhttps://indico.math.cnrs.fr/event/4274/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/4274/
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