Dynamical Systems Seminar - Wild holomorphic dynamical systems

Speaker: Sebastien Biebler (Universite de Paris VII, MSRI)

Abstract:  In the 60s, in a mathematical optimistic movement aiming to describe a typical dynamical system, Smale conjectured the density of uniform hyperbolicity in the space of C^r-diffeomorphisms f of a compact manifold M. In 1974, a student of Smale, Newhouse, discovered an extremely complicated new phenomenon, resulting in an obstruction to Smale's conjecture. Specifically, Newhouse showed the existence of (nonempty) open sets U  of C^2-diffeomorphisms of a surface M such that a generic map f in U has infinitely many attracting periodic points. In particular, the statistical behavior of such systems can not be described in a satisfying way with a finite number of measures.
In this talk, I will first define precisely the Newhouse phenomenon. Then, I will discuss a joint work with Pierre Berger whose proof is based on the Newhouse phenomenon. We show that there exist polynomial automorphisms of C^2 with a wandering Fatou component. This result contrasts with a celebrated theorem of Sullivan who proved in the 80s that any rational map of the Riemann sphere does not have such wandering Fatou components. We also study the statistical behaviors of orbits of points inside the wandering component, and we show that it is very difficult to describe, namely historic with high emergence.