$\mathbb{C}^{2}$自同构中的有孔抛物柱面
Punctured parabolic cylinders in automorphisms of $\mathbb{C}^{2}$
摘要：
我们证明了存在的自同构$F$的$\mathbb{C}^{2}$与一个非经常性的Fatou组件$Ω$双全纯$\mathbb{C}\times\mathbb{C}^{*}$，这是盆地吸引区域力不变的整个曲线上，$F$作为一个不合理的旋转。我们进一步证明了可以选择$Ω\to\mathbb{C}\times\mathbb{C}^{*}$，使得它将$F$共轭到平移$(z,w)\mapsto(z+1,w)$，使$Ω$成为最近由L.~ Boc Thaler，F.~ Bracci和H.~彼得斯$F$和$Ω$是通过将$\mathbb{C}^{2}$的一个自同构的一个不动点与吸引到该不动点的一个具有相同双全纯类型的Fatou分量爆破而得到的，该不动点由F.~布拉奇，J.~ Raissy和B.~ Stensønes.关键的一步是应用一个合适的李代数的密度性质来证明他们工作中的自同构可以被选择，使得它固定一个坐标轴。然后，我们可以从爆破中移除该轴的适当变换，以获得与$\mathbb{C}^{2}$双全纯的爆破的$F$稳定子集。因此，我们可以将$F$解释为$\mathbb{C}^{2}$的自同构。
英文摘要：
We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $Ω$ 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 $Ω\to\mathbb{C}\times\mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto(z+1,w)$, making $Ω$ a parabolic cylinder as recently defined by L.~Boc Thaler, F.~Bracci and H.~Peters. $F$ and $Ω$ 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 $\mathbb{C}^{2}$.
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