The Yoccoz Combinatorial Analytic Invariant
The Yoccoz Combinatorial Analytic Invariant
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DOI：
10.1090/fic/053/07
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
C. Petersen;P. Roesch
中科院分区：
文献类型：
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作者：
C. Petersen;P. Roesch
在本文中，我们开发了一个组合解析编码的Mandelbrot集M。编码是隐含在Yoccoz'证明局部连通性的M在任何Yoccoz参数，即任何至多可重正化的参数，所有的周期轨道是re-pelling。使用这种编码，我们定义了一个显式的组合分析模型空间，它是足够抽象的，它可以作为一个中间人来证明其他集合，如抛物的曼德尔-布罗特集M <$具有相同的组合结构。作为一个直接应用，我们在这里使用组合分析模型来重新证明M的并矢脉是弧，更一般地，任何两个Yoccoz参数都由M中唯一的直纹（在Douady-Hubbard意义上）弧连接。
In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz'proof of local connectivity of M at any Yoccoz parameter, ie any at most finitely renormalizable parameter for which all periodic orbits are re-pelling. Using this encoding we define an explicit combinatorial ana-lytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic¹ Mandel-brot set M₁ has the same combinatorial structure as M. As an im-mediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M.