Punctured parabolic cylinders in automorphisms of ℂ2

We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $\Omega $ biholomorphic to $\mathbb{C}\times \mathbb{C}^{∗}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $\Omega \to \mathbb{C}\times \mathbb{C}^{∗}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto (z+1,w)$, making $\Omega $ a parabolic cylinder as recently defined by L. Boc Thaler, F. Bracci, and H. Peters. $F$ and $\Omega $ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}^{2}$ with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F. Bracci, J. Raissy, and B. Stensønes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an $F$-stable subset of the blow-up that is biholomorphic to $\mathbb{C}^{2}$. Thus, we can interpret $F$ as an automorphism of $\mathbb{C}^{2}$.