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

Abstract

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 ${\mathbb{C}}^k$.

this paper was previously entitled "Automorphisms of ${\mathbb{C}}^2$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb{C}\times {\mathbb{C}}^{\ast }$". this is the final version, accepted for publication in Journal Eur. Math. Soc. (JEMS)

References in corpus (1)

• Runge tubes in Stein manifolds with the density property

Cited by in corpus (3)

• A transcendental Hénon map with an oscillating wandering Short ${\mathbb{C}}^2$
• Automorphisms of ${\mathbb{C}}^2$ with parabolic cylinders
• Invariant escaping Fatou components with two rank 1 limit functions for automorphisms of ${\mathbb{C}}^2$