周期轨道、外部光线与曼德布洛集：一个解释性的说明
Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account
摘要：
Douady和Hubbard研究Mandelbrot集$M$的一个关键点是定理，即$M$中的每一个抛物点$c\ne 1/4$是恰好两条具有角度的外部光线的着陆点，这两条光线在加倍作用下是周期性的。本文将试图提供一个证明这一结果和它的一些后果，这依赖于尽可能多的初等组合，而不是更困难的分析。它的灵感来自Schleicher最近论文的第2部分（另见Stony Brook IMS预印本1994/19，与E。Lau），其中包含非常大量的简化Douady-Hubbard证明与一个更紧凑的论点，并强烈推荐。这里给出的证明是相当不同的Schleicher，是基于组合研究的角度外射线的朱利亚集土地上的周期轨道。本文中的结果大多是众所周知的;与Douady和Hubbard的工作有特别强的重叠。唯一声称的独创性是在强调，和组织的证据。
英文摘要：
A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for exactly two external rays with angle which are periodic under doubling. This note will try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard proofs with a much more compact argument, and is highly recommended. The proofs given here are rather different from those of Schleicher, and are based on a combinatorial study of the angles of external rays for the Julia set which land on periodic orbits. The results in this paper are mostly well known; there is a particularly strong overlap with the work of Douady and Hubbard. The only claim to originality is in emphasis, and the organization of the proofs.
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