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BEGIN:VEVENT
SUMMARY:School - Limit Theorems for dynamical systems
DTSTART:20130527T080000
DTEND:20130531T180000
DTSTAMP:20260406T172855Z
UID:5c84497662a5a2b0080d645c71394cdaf9c468db263d67ca827fa737
CATEGORIES:Conferences - Seminars
DESCRIPTION:FORNI Giovanni (University of Maryland).\nGOUËZEL Sébastien 
 (Université de Rennes 1).\nMARKLOF Jens (University of Bristol).\nJ. Mark
 lof (University of Bristol)\nKinetic limits of dynamical systems\nSince th
 e pioneering work of Maxwell and Boltzmann in the 1860s and 1870s\, a majo
 r challenge in mathematical physics has been the derivation of macroscopic
  evolution equations from the fundamental microscopic laws of classical or
  quantum mechanics. Macroscopic transport equations lie at the heart of ma
 ny important physical theories\, including fluid dynamics\, condensed matt
 er theory and nuclear physics. The rigorous derivation of macroscopic tran
 sport equations is thus not only a conceptual exercise that establishes th
 eir consistency with the fundamental laws of physics: the possibility of f
 inding deviations and corrections to classical evolution equations makes t
 his subject both intellectually exciting and relevant in practical applica
 tions.\nThe plan of the these lectures is to develop a renormalization tec
 hnique that will allow us to derive transport equations for the kinetic li
 mits of certain dynamical systems\, including the Lorentz gas and kicked H
 amiltonians (linked twist maps). The technique uses the ergodic theory of 
 flows on homogeneous spaces (homogeneous flows for short)\, and is based o
 n joint with Andreas Strömbergsson\, Uppsala. I will explain the basic st
 eps of the renormalisation approach\, give a gentle introduction to the er
 godic theory of homogeneous flows\, and discuss key properties of the macr
 oscopic transport equations that emerge in the kinetic limit. The lectures
  are aimed at a broad mathematical audience.\nG. Forni (University of Mary
 land)\nLimit theorems for classical horocycle flows\nIn these lectures we 
 describe joint results with L. Flaminio and A. Bufetov on the deviation of
  ergodic averages and limit distributions of ergodic integrals of smooth f
 unctions for horocycle flows on the unit tangent bundle of compact surface
 s of constant negative curvature. The classical horocyle flow is a parabol
 ic (zero entropy) renormalizable dynamical system\, it is uniquely ergodic
 \, mixing of all orders\, with nearly but not quite integrable decay of co
 rrelations. It is the simplest parabolic renormalizable system for which t
 he study of deviation of ergodic averages and limit distributions can be c
 arried out in some detail by tools of harmonic analysis\, namely of the th
 eory of unitary representations of the group PSL(2\,R).\nThe results we wi
 ll describe should be considered as a model for generalizations to other s
 ystems of the same kind such as interval exchange tranformations and flows
  on surfaces (carried out by Bufetov)\, 2-step nilflows or horospherical f
 oliations of geodesic flows on manifolds of negative curvature. The lectur
 es will be based mostly on the following two papers:\nL. Flaminio and G. F
 orni: Invariant distributions and time averages for horocycle flows. Duke 
 Math. J. Volume 119\, Number 3 (2003)\, 465-526.\nA. Bufetov and G. Forni:
  Limit Theorems for Horocycle Flows\, available at arxiv.org/pdf/1104.4502
 \nS. Gouezel (Universite de Rennes)\nLimit theorems in dynamical systems u
 sing the spectral method\nWhile martingale arguments are often convenient 
 to prove limit theorems in dynamical systems\, some classes of problems an
 d some classes of systems are not amenable to such arguments. I will expla
 in another method\, the so-called Nagaev-Guivarc'h spectral method\, that 
 is often fruitful is such situations. Starting from the simplest example (
 the central limit theorem for interval maps)\, we will also describe more 
 recent applications\, for instance to the convergence towards stable distr
 ibutions\, or to the almost sure invariance principle.
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STATUS:CONFIRMED
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