Nonlinear Dynamics, Ergodic Theory and Renormalization

The workshop is aimed at the overlap of Mathematics and Physics, where the subjects of nonlinear dynamics, ergodic theory and renormalisation meet. This is an area of science with a great development, where participants can learn a lot from each other. A major theme is the mathematical characterization of chaotic dynamics, which up to now only is succesful in the lower dimensional setting, but where currently methods are being developed for generalization to higher dimensions. In this program ergodic theory and renormalization theory are important tools, which partially have to be developed from start, but also often are adapted from other area’s in mathematical physics. We expect that during the workshop there will be a lot of cross fertilization between various groups of mathematicians and physicists in related area’s.

Our main interest focuses on two area’s where physicists and mathematicians may interact strongly, namely ‘ergodic theory’ and ‘renormalization’.

ERGODIC THEORY: This concerns finite dimensional invariant measures with concepts like decay of correlation and stochastic stability, and their meaning for physical theories, e.g., concerning the ergodic hypothesis (Van Beijeren, Posch, Liverani, Pujals, Nowicki).

RENORMALIZATION: Correspondences and differences in the various uses of renormalization techniques in finite dimensional dynamics, fluid dynamics and statistical mechanics (Kupiainen, Bricmont, Jona-Lasinio, Martinelli, Lyubich, Eckmann, Coullet, Tresser).

Concrete Problems

To fix thoughts we formulate a few concrete examples that belong to one or both of these area’s.

KAM theory: Does the relative measure of KAM tori tend to _ as the number of degrees of freedom tends to infinity? This subject concerns the relationship of finite dimensional ergodicity with the ergodic hypothesis of statistical mechanics (ergodicity - Van Enter, Van Beijeren).

Localization and spectra of Schrödinger operator (renormalisation - Janssen, Avron, Avila, Krikorian, Yoccoz, Marmi).

ARNOLD-AVEZ conjecture: For rather general Hamiltonian systems show that areas of positive measure exist with positive Lyapunov exponent (ergodicity - Sinai).

STRANGE ATTRACTORS: General phenomena accompanying strange- or chaotic attractors in dimension larger than two (both ergodicity and renormalization - Takens, Martens).

Up to now there is a good theory of hyperbolic attractors in terms of topological stability, ergodic theory and the thermodynamic formalism. For non-hyperbolic strange attractors there is only one main general result: the Oseledic theory of characteristic exponents and the corresponding invariant manifolds. In dimension two these non-hyperbolic atttractors are now much better understood. For example it is known that whenever a diffeomorphism in dimension two has a non-hyperbolic strange attractor, then, by an arbitrarily small perturbation one can obtain a dynamical system

1. with many (even infinitely many) periodic attractors, or

2. with a homoclinic tangency, or

3. with a Hénon-like strange attractor.

And probably even a systen with all three phenomena at the same time.

This can be seen as an analogue of the well-known fact that in each hyperbolic strange attractor there are many subsystems of horseshoe type.

In dimension larger than two the situation is completely different: (2) may still hold, but (1) and (3) are definitely false. At this moment there are examples of new phenomena in higher dimensions, but we are still far from a description of the phenomena which are always present in or near nonhyperbolic strange attractors in these higher dimensions. Many present publications can be seen as steps in the direction of the solution of this general problem.

Overview talks on KAM theory (Broer, Yoccoz), on the Hénon family (Simo) and on the Brazilian program (Viana) will needed.