Speaker
Jacob Palis
Title
Open questions leading to a global perspective in dynamics
Abstract
We will address one of the most challenging and central problems in dynamical systems, meaning flows, diffeomophisms or, more generally, transformations, defined on a compact manifold without boundary or an interval on the real line: can we describe the behavior in the long run of typical trajectories for typical systems? Poincaré was probably the first to point in this direction and stress its importance. We shall consider finite-dimensional C∞, Cr with r≥1, parameterized families of dynamics endowed with the C∞, Cr topology. The concept of “typical” in our context is taken in terms of Lebesgue probability both in parameter and phase spaces. Our purpose is to discuss a conjecture stating that for a typical dynamical system, almost all trajectories have only finitely many choices, of (transitive) attractors, where to accumulate upon in the future. Interrelated conjectures will also be discussed. If any of the attractors is not reduced to a fixed or periodic trajectory, the systems is called chaotic and uncertainty is measured by the diameter of the attractors.