单叶多项式与Hubbard树
Univalent Polynomials and Hubbard Trees
摘要：
研究了次数为$d+1$的有理函数$f$，使得$f$在外单位圆上是单叶的，且单位圆在$f$下的象具有最大尖点数（$d+1$）和最大双点数$(d-2)$.我们引入了一个双角度树与任何这样的$f$。证明了任意双角树都可由这样一个$f$实现，而且$f$本质上由它的双角树唯一确定.这种组合分类是用来表明，这样的$f$是在自然的1：1的对应关系与反全纯多项式的程度$d$与$d-1$不同的，固定的临界点（分类的哈伯德树）。
英文摘要：
We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such $f$ are in natural 1:1 correspondence with anti-holomorphic polynomials of degree $d$ with $d-1$ distinct, fixed critical points (classified by their Hubbard trees).
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