The combinatorial Mandelbrot set as the quotient of the space of geolaminations
The combinatorial Mandelbrot set as the quotient of the space of geolaminations
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组合 Mandelbrot 设置为地质层压空间的商
DOI：
10.1090/conm/669/13422
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Timorin Vladlen
中科院分区：
文献类型：
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作者：
B. Alexander;Oversteegen Lex;Ptacek Ross;Timorin Vladlen
我们解释组合的Mandelbrot集的\它{二次叠层}（等价关系$\sim$下的单位圆不变$\sigma_2 $）。对于每一个叠层，我们关联一个特定的{\em geolamination}（集合$\mathcal{L}_\sim$的圆的点和$\sim$-等价类的凸包的边），使得它们的集合的闭包是一个具有豪斯多夫度量的紧度量空间。两个这样的地质分层被认为是{\em小调等价}，如果它们的{\em小调}（它们最长和弦的图像）相交。证明了该拓扑空间的商空间与组合Mandelbrot集的边界同胚。对于这些地质分层的每个等价类，我们关联一个唯一的分层及其拓扑多项式，以便这种解释可以被视为一种赋予所有二次拓扑多项式空间一个合适的拓扑结构的方法。
We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations $\sim$ on the unit circle invariant under $\sigma_2$). To each lamination we associate a particular {\em geolamination} (the collection $\mathcal{L}_\sim$ of points of the circle and edges of convex hulls of $\sim$-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be {\em minor equivalent} if their {\em minors} (images of their longest chords) intersect. We show that the corresponding quotient space of this topological space is homeomorphic to the boundary of the combinatorial Mandelbrot set. To each equivalence class of these geolaminations we associate a unique lamination and its topological polynomial so that this interpretation can be viewed as a way to endow the space of all quadratic topological polynomials with a suitable topology.