拟正则 Mandelbrot 集的等势线与折叠

中文摘要 AI 辅助

众所周知，Douady 和 Hubbard 通过利用 Böttcher 坐标构造补集的均匀化全纯映射，证明了 Mandelbrot 集是连通的。这一策略在拟正则 Mandelbrot 集的设定中有明显的类比。在本文中，我们证明了 Douady-Hubbard 均匀化映射的自然推广在无穷远邻域内是拟共形的，事实上，在拟正则 Mandelbrot 集的某个邻域之外也是拟共形的。另一方面，我们还证明了该均匀化映射在某些参数下可以反转方向，这迫使它产生折叠，特别是阻止了它的单射性。这一特征在全纯设定中不会发生。因此，拟正则 Mandelbrot 集的连通性无法沿着 Douady-Hubbard 路线建立，并且仍然是一个未解决的问题。

英文摘要

It is well-known that Douady and Hubbard proved that the Mandelbrot set is connected by constructing a uniformizing holomorphic map for its complement from the Böttcher coordinates. This strategy has a clear analogue in the setting of quasiregular Mandelbrot sets. In this paper, we show that the natural generalization of the Douady-Hubbard uniformizing map is quasiconformal in a neighbourhood of infinity, as well as, in fact, outside a certain neighbourhood of the quasiregular Mandelbrot set. On the other hand, we also show that this uniformizing map can reverse orientation at some parameters, which forces it to have folds and, in particular, prevents it from being injective. This feature cannot happen in the holomorphic setting. The connectivity of the quasiregular Mandelbrot sets therefore cannot be established along the Douady-Hubbard route, and remains open.

发表机构

• Northern Illinois University（北伊利诺伊大学）

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