Automorphisms of $\mathbb C^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$

Filippo Bracci (1) , Jasmin Raissy (2) , Berit Stensønes (3)

1 DIPMAT - Dipartimento di Matematica [Roma II]
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 NTNU - Norwegian University of Science and Technology [Trondheim]

We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such a Fatou component also avoids k analytic discs intersecting transversally at the fixed point. As a corollary, we obtain a Runge copy of $\mathbb C\times (\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.

Mots clés: hyperbolic distance; Fatou components; Runge domains; holomorphic dynamics

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Filippo Bracci, Jasmin Raissy, Berit Stensønes. Automorphisms of $\mathbb C^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$. Journal of the European Mathematical Society, In press. ⟨hal-01610350v2⟩