拉直映射II的组合学和拓扑学：不连续性
Combinatorics and topology of straightening maps II: Discontinuity
摘要：
我们继续研究$d \ge 3$次多项式族的拉直映射。拉直映射的概念最初是由Douady和Hubbard为了研究Mandelbrot集的类多项式重整化与自相似性之间的关系而引入的。在二次情形下，拉直映射总是连续的，这是证明Mandelbrot集本身有小副本的关键步骤之一。另一方面，对于更高次的情形，我们不具有这样好的自相似性质：正如Douady和Hubbard所期望的具有不连续拉直映射的类X族的例子，我们证明了拉直映射是不连续的，除非它是不相交的。
英文摘要：
We continue the study of straightening maps for the family of polynomials of degree $d \ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and self-similarity of the Mandelbrot set. In the quadratic case, straightening maps are always continuous, and this is one of the critical steps to prove the Mandelbrot set has small copies in itself. On the other hand, for higher degree case, we do not have such a nice self-similar property: As expected from an example of a cubic-like family with discontinuous straightening map by Douady and Hubbard, we prove that the straightening map is discontinuous unless it is of disjoint type.
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