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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. [Back] |