*(University of Bristol)*

**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 StrÃ¶mbergsson, 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

at arxiv.org/pdf/1104.4502

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
of

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.