Laminational Models for Some Spaces of Polynomials of Any Degree
Laminational Models for Some Spaces of
Polynomials of Any Degree
复制标题
任意次数多项式的某些空间的层状模型
DOI：
10.1090/memo/1288
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
V. Timorin
中科院分区：
文献类型：
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作者：
A. Blokh;L. Oversteegen;R. Ptacek;V. Timorin
所谓的Mandelbrot集的“夹盘”模型是由A.~Douady、J.~H.~Hubbard和W.~P.~瑟斯顿提出的。它可以用测地线分层的语言来描述。组合模型是单位圆盘在等价关系下的商空间，松散地说，该等价关系“挤压”平面中的圆盘(由此模型的名称)。该模型的意义尤其在于这个商是平面的，因此很容易被可视化。Mandelbrot集实际上同胚于该模型的猜想等价于著名的MLC猜想，即Mandelbrot集是局部连通的。
对于高次多项式的参数空间，没有已知的组合模型。一个可能的原因可能是，MLC猜想的更高程度的类比已知是错误的。我们研究测地线分层在多大程度上由其临界集的位置决定，以及当临界集的不同选择导致实质上相同的分层时。这产生了各种参数空间的分层模型，类似于Mandelbrot集的“挤压圆盘”模型。
The so-called "pinched disk" model of the Mandelbrot set is due to A.~Douady, J.~H.~Hubbard and W.~P.~Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, "pinches" the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected.
For parameter spaces of higher degree polynomials no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the "pinched disk" model of the Mandelbrot set.