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}$|⁠.

中文翻译：

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morph ²自同构的穿刺抛物线形圆柱

我们展示了同构| $ F $ |的存在。的| $ \ mathbb {C} ^ {2} $ | 具有非经常性法图成分的| $ \ Omega $ | 双全纯以| $ \ mathbb {C} \次\ mathbb {C} ^ {*} $ | 那是对| $ F $ |不变的整个曲线的吸引的盆地。充当非理性的轮换。我们进一步证明| $ \ Omega \ to \ mathbb {C} \ times \ mathbb {C} ^ {*} $ | 可以选择为使得| $ F $ |共轭 到翻译| $（z，w）\ mapsto（z + 1，w）$ |⁠，使| $ \ Omega $ | 最近由L. Boc Thaler，F。Bracci和H. Peters定义的抛物柱面。| $ F $ | 和| $ \ Omega $ | 通过炸毁| $ \ mathbb {C} ^ {2} $ | F. Bracci，J。Raissy和B.Stensønes建立的具有相同双全型类型的Fatou组分被吸引到该固定点。至关重要的一步是应用合适的李代数的密度属性，以表明可以选择其工作中的自同构以固定坐标轴。然后，我们可以从爆炸中删除该轴的正确变换，以获得| $ F $ |。是| $ \ mathbb {C} ^ {2} $ |⁠的全变态的爆炸的稳定子集。因此，我们可以解释| $ F $ | 作为| $ \ mathbb {C} ^ {2} $ |⁠的同构。