Kinetic limits of dynamical systems
Since the pioneering work of Maxwell and Boltzmann in the 1860s and 1870s, a major 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 many important physical theories, including fluid dynamics, condensed matter theory and nuclear physics. The rigorous derivation of macroscopic transport equations is thus not only a conceptual exercise that establishes their consistency with the fundamental laws of physics: the possibility of finding deviations and corrections to classical evolution equations makes this subject both intellectually exciting and relevant in practical applications.
The plan of the these lectures is to develop a renormalization technique that will allow us to derive transport equations for the kinetic limits of certain dynamical systems, including the Lorentz gas and kicked Hamiltonians (linked twist maps). The technique uses the ergodic theory of flows on homogeneous spaces (homogeneous flows for short), and is based on joint with Andreas Strömbergsson, Uppsala. I will explain the basic steps of the renormalisation approach, give a gentle introduction to the ergodic theory of homogeneous flows, and discuss key properties of the macroscopic transport equations that emerge in the kinetic limit. The lectures are aimed at a broad mathematical audience.
G. Forni (University of Maryland)
Limit theorems for classical horocycle flows
In 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 functions for horocycle flows on the unit
tangent bundle of compact surfaces of constant negative curvature. The
classical horocyle flow is a parabolic (zero entropy) renormalizable
dynamical system, it is uniquely ergodic, mixing of all orders, with
nearly but not quite
integrable decay of correlations. It is the simplest parabolic renormalizable system for which the study of deviation of ergodic averages and limit
distributions can be carried out in some detail by tools of harmonic analysis, namely of the theory of unitary representations of the group PSL(2,R).
The results we will describe should be considered as a model for generalizations to other systems of the same kind such as interval exchange tranformations and flows on surfaces (carried out by Bufetov), 2-step nilflows or horospherical foliations of geodesic flows on manifolds of negative curvature. The lectures will be based mostly on the following two papers:
L. Flaminio and G. Forni: Invariant distributions and time averages for
horocycle flows. Duke Math. J. Volume 119, Number 3 (2003), 465-526.
A. Bufetov and G. Forni: Limit Theorems for Horocycle Flows, available
S. Gouezel (Universite de Rennes)
Limit theorems in dynamical systems using the spectral method
While martingale arguments are often convenient to prove limit
theorems in dynamical systems, some classes of problems and some classes
systems are not amenable to such arguments. I will explain 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 distributions, or to the almost sure invariance principle.