Title:

Emergence of wild dynamics

Abstract:

The dynamical study of a diffeomorphism of a manifold is usually split into two parts: the stable part in which (typical) nearby points remain close during their orbit, and the chaotic part in which (typical) nearby points separate exponentially fast during their orbit. One could believe that the stable part of the dynamics is simpler than the chaotic one, and that it would be possible to present a study of these two parts by modeling the statistical behavior of most of the points using a small number of probability measures. To the contrary we will see that the stable part of the dynamics can be statistically extremely complex: its statistical behavior need at least an infinitely dimensional space of probability measures to be described. This complexity will be quantified by a new invariant: the emergence. We will see that such dynamics exist not only among differentiable dynamics, but also among symplectic or in real and complex analytic ones.