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. As a corollary, we obtain a Runge copy of $\mathbb{C}\times ({\mathbb{C}}^{\ast }{)}^{k-1}$ in ${\mathbb{C}}^k$. The constructed Fatou component also avoids $k$ analytic discs intersecting transversally at the fixed point.