后奇异有限$(p,q)$ -指数映射的有界几何与刻画
Bounded Geometry and Characterization of post-singularly Finite $(p,q)$-Exponential Maps
摘要：
本文定义了平面上分支覆盖映射的拓扑类{$(p,q)$型\em拓扑指数映射}，记为$\TE_{p,q}$，其中$p\geq 0$和$q\geq 1$ .我们遵循\cite{Ji}中给出的框架来研究组合地表征整个映射$P e^{Q}$的问题，其中$P$是次数为$p$的多项式，$Q$是次数为$q$的多项式，使用由Thurston定义的{ \em迭代方案}和{ \em有界几何条件}。我们首先证明了具有有限后奇异集的元素$f \in {\TE}_{p,q}$组合等价于整映射$P e^{Q}$当且仅当它具有紧的有界几何。因此，为了完成刻画，我们只需要检查有界几何实际上蕴涵紧性。我们展示了一些$f\in \TE_{p,1}$，$p\geq 1$。本文的主要结果是：只有一个非零单断点$c$使得$c$是周期的或$c$和$f(c)$都不是周期的后奇异有限映射$f$在$\TE_{0,1}$中或后奇异有限映射$f$在$\TE_{p,1}$中，$p\geq 1$组合等价于形式为$e^{λz}$或形式为$ αz^{p}e^{λz}$的后奇异有限整映射，其中分别为$α=(-λ/p)^{p}e^{- λ(-p/λ)^{p}}$，当且仅当它具有有界几何。这是在这个方向上的第一个结果，一族超越全纯映射的临界点。
英文摘要：
In this paper we define a topological class of branched covering maps of the plane called {\em topological exponential maps of type $(p,q)$} and denoted by $\TE_{p,q}$, where $p\geq 0$ and $q\geq 1$. We follow the framework given in \cite{Ji} to study the problem of combinatorially characterizing an entire map $P e^{Q}$, where $P$ is a polynomial of degree $p$ and $Q$ is a polynomial of degree $q$ using an {\em iteration scheme defined by Thurston} and a {\em bounded geometry condition}. We first show that an element $f \in {\TE}_{p,q}$ with finite post-singular set is combinatorially equivalent to an entire map $P e^{Q}$ if and only if it has bounded geometry with compactness. Thus to complete the characterization, we only need to check that the bounded geometry actually implies compactness. We show this for some $f\in \TE_{p,1}$, $p\geq 1$. Our main result in this paper is that a post-singularly finite map $f$ in $\TE_{0,1}$ or a post-singularly finite map $f$ in $\TE_{p,1}$, $p\geq 1$, with only one non-zero simple breanch point $c$ such that either $c$ is periodic or $c$ and $f(c)$ are both not periodic, is combinatorially equivalent to a post-singularly finite entire map of either the form $e^{λz}$ or the form $ αz^{p}e^{λz}$, where $α=(-λ/p)^{p}e^{- λ(-p/λ)^{p}}$, respectively, if and only if it has bounded geometry. This is the first result in this direction for a family of transcendental holomorphic maps with critical points.
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