Abstract:

As discovered by Poincare in the end of the 19th century, even small perturbations of very regular dynamical systems may display chaotic features, due to complicated interactions near a homoclinic point. In the 1960s, Smale attempted to understand such dynamics in term of a stable model, the horseshoe, but this was too optimistic. Indeed, Newhouse showed that even in only two dimensions, a homoclinic bifurcation gives rise to particular wild dynamics, such as the generic presence of infinitely many attractors. This Newhouse phenomenon is associated to a renormalization mechanism, but also with particular geometric properties of some fractal sets within a Smale horseshoe. When considering two dimensional complex dynamics those fractal sets become much more beautiful but unfortunately also more difficult to handle.

Speaker:

Artur Ávila is a Brazilian mathematician who has made deep contributionsto dynamical systems theory which has transformed the subject. His work has been honoured with numerous prizes including a Fields Medal in 2014.