Seminar

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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 the 70s, Newhouse discovered an extremely complicated new phenomenon, resulting in an obstruction to Smale's conjecture. Specifically, he 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 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.

Joint work with Pierre Berger, CNRS, Sorbonne University.