Abstract

In these lectures we consider how algebraic properties of discrete subgroups of Lie groups restrict the possible actions of those groups on surfaces. The results show a strong parallel between the possible actions of such a group on the circle S¹ and the measure preserving actions on surfaces.

Our aim is the study of the (non)-existence of actions of lattices in a large class of non-compact Lie groups on surfaces. A definitive analysis of the analogous question for actions on S¹ was carried out by É. Ghys. Our approach is topological and insofar as possible we try to isolate properties of a group which provide the tools necessary for our analysis. The two key properties we consider are almost simplicity and the existence of a distortion element. Both will be defined and described in the lectures.

Our techniques are almost all from low dimensional dynamics. But we are interested in how algebraic properties of a group, commutativity, nilpotence, etc. affect the possible kinds of dynamics which can occur. For most of the results we will consider groups of diffeomorphisms which preserve an Borel probability measure.

Abstract

The main goal is to show how the numerical approximation of invariant objects can be used to study a given dynamical system.

In the course we will first focus on several concrete problems and we will show how the computation of invariant manifolds appear as a natural tool to understand the geometry of phase space. Then, we will introduce several numerical tools and we will discuss how to implement them in a computer. Finally, the results will be used to describe the considered phenomena.

The examples come from Hamiltonian dynamics, celestial mechanics, dissipative low dimensional systems and PDEs. The considered tools are the analysis of bifurcations, the computation of invariant manifolds (invariant tori, stable and unstable manifolds, etc.) and the estimation of Lyapunov exponents.

Abstract

Interval exchange transformations have been a continuous source of beautiful questions (and beautiful answers), not the least because they relate very closely to many other topics, such as poligonal billiards, measured foliations, Teichmuller geodesic flows, flat surfaces, renormalization theory, continued fraction expansions, and yet many others.

The Zorich-Kontsevich conjecture, formulated in the mid- nineties, proposes a surprisingly detailed description of the behavior of typical interval exchange transformations in terms of an "asymptoctic flag". The conjecture was recently proved by Avila and myself, and the proof also contains very important partial progress obtained a couple of years ago by Forni.

The course will be a self contained introduction to this field, meant to convey the flavor of the main objects and their relationships, as well as some outline of the proof of the conjecture.