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

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