A two-dimensional polynomial map with a wandering Fatou component

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Jasmin Raissy
Université Paul Sabatier, Toulouse

The Fatou set of a holomorphic endomorphism of a complex manifold is the largest open set where the family iterates of the map form a normal family, and a Fatou component is a connected component of the Fatou set. In dimension one, Sullivan's Non Wandering Domain Theorem asserts that every Fatou component of a rational map is eventually periodic. Several classes of counter-examples have been found and studied for entire transcendental function in dimension one, but the question of the existence wandering Fatou components for polynomial endomorphisms in higher dimension remained open.
We show, using techniques of parabolic bifurcation, that there exist polynomial endomorphisms of $\mathbb{C}^2$ with a wandering Fatou component. These maps are polynomial skew-products, and can be chosen to extend to holomorphic endomorphisms of the complex projective space. (Joint work with M. Astorg, X. Buff, R. Dujardin and H. Peters)