We study the iteration of transcendental self-maps of $\mathcal{C}^*:\=\mathcal{C}\{0}$, that is, holomorphic functions $\fnof:\mathcal{C}^*:\rarr\mathcal{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0},\infin$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathcal{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

中文翻译：

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穿孔平面先验自图的逃逸法头分量

我们研究了先验自映射的迭代$\mathcal{C}^*:\=\mathcal{C}\{0}$，即全纯函数$\fnof:\mathcal{C}^*:\rarr\mathcal{C}^*$其中零和无穷大都是本质奇点。我们使用近似理论来构造这个类中的函数，其中包含逃逸的 Fatou 组件，包括流浪域和贝克域，它们累积到$\{0},\infin$以任何可能的方式在迭代下。我们还给出了先验自我映射的第一个显式例子$\mathcal{C}^*$与贝克域和流浪域。在这样做的过程中，我们开发了一个函数具有一个简单连接的逃逸漫游域的充分条件。最后，我们注意到我们的结果还提供了带有转义 Fatou 组件的整个函数的新示例。