Exponential scattering of trajectories and its hydrodynamical applications

Abstract

At the beginning of the 1960s in the theory of dynamical systems with finitely many degrees of freedom some remarkable events occurred: the stability of exponential divergence of trajectories in phase space or on a subset of it attracting neighbouring trajectorieswas discovered. The fundamental importance of these discoveries for the description of turbulence-type phenomena both in finite-dimensional and infinite-dimensional systems was understood fairly soon. However, this idea became widely accepted only in the seventies after Ruelle and Takens invented the term “attractor”, which rapidly became fashionable, and took the whole area out of the restraining framework of rigorous theorems (of Anosov, Sinai, Smale, and others) into the vast terrain of numerical and physical experiments.

In this lecture, I will talk on three interrelated directions of rigorous mathematical investigation, resulting from the interaction of ideas in the modern theory of finite-dimensional dynamical systems and hydrodynamics: upper estimates of the dimension of attractors, applications of the geometry of infinite-dimensional Lie groups in the hydrodynamics of an ideal fluid, and certain questions relating to the problem of the stationary kinematic dynamo in magneto-hydrodynamics.