关于指数映射族逃逸射线聚点集与奇异扰动有理映射族Julia集的研究

发布时间：2018-01-21 10:54

  本文关键词： 指数映射族 不着陆 逃逸射线 聚点集 不可分割的连续统 奇异扰动 逃逸 拟圆周 Cantor圆周 Sierpinski地毯 无穷连通Fatou分支 Herman环　出处：《南京大学》2016年博士论文　论文类型：学位论文

【摘要】：这篇博士论文主要包含以下两部分：第一部分是关于指数映射族逃逸射线聚点集的研究.作为超越整函数动力系统的典型研究对象,指数映射族的动力系统一直备受关注.其中一个重要的研究课题就是对其不着陆逃逸射线的研究.在这项研究之前,人们所发现的指数映射族的不着陆的逃逸射线都聚属于聚点集无界的类型,更精确的说,它们的聚点集都是复平面中无界的不可分割的连续统,并且必须包含某条逃逸射线的全部作为其聚点集的一部分.在本文中,作者对指数映射族构造出了这样的逃逸射线：它们的聚点集是复平面中的紧集.更进一步,作者通过引入折叠模型,构造出了三种新类型的逃逸射线.对每一条这样的逃逸射线,作者定义了与之相关的一个返回序列.依据这个返回序列的组合特征,作者对聚点集的拓扑做了如下三种分类：(1)包含部分逃逸射线的不可分割的连续统；(2)与逃逸射线互不相交的不可分割的连续统；(3)Jordan弧.第二部分是关于一族奇异扰动有理映射上的动力系统.当作为Pn(z)=zn的扰动时,我们构造的函数族所扰动出来的Julia集是Cantor圆周,但是此Cantor圆周上的动力系统与传统的McMullen映射族所得到的Cantor圆周上的动力系统却不是拓扑共轭的.一方面,作者研究了此函数族在自由临界点逃逸到0或者∞超吸引域的情形下(双曲情形),按其逃逸到0或者∞超吸引域时的迭代次数,对其Julia集所有可能的情形进行了分类.这里得到的Julia集可以分为拟圆周,Cantor圆周,Sierpinski地毯和退化的Sierpinski地毯共四种情形.我们可以看出它此时具有非常丰富的动力学行为.并且,在每种情况下,作者还给出了具体的参数来说明相应的情况的确会发生.特别地,作者给出了此情形下0和∞超吸引域边界的正则性,证明了在这种情况下∞的直接超吸引域的边界一定是一个拟圆周.对于Julia集是拟圆周的情形,作者给出了当参数是实数时的精确范围.对于Cantor圆周情形,作者给出了Cantor圆周存在性关于映射度的一个充要条件.另一方面,作者还研究了此函数族在所有情形下Julia集的连通性.通过讨论其自由临界轨道是否逃逸到0或者∞的超吸引域中,作者给出了其Julia集不连通的充要条件：其Julia集不连通当且仅当它是Cantor圆周.这等价于这个函数族有一个临界值包含在0或者∞的超吸引域中,而其相应的临界点却不在其中.这个结果可以看作是经典二次多项式的Julia集连通性相关结论的一种类比.
[Abstract]:This dissertation mainly consists of the following two parts: the first part is about the study of exponential mapping family escape ray accumulation point set, as a typical research object of transcendental whole function dynamic system. The dynamical system of exponential mapping family has been paid much attention. One of the important research topics is the study of its non-landing escape ray. It is found that the untouched escape rays of exponential mapping family belong to the unbounded type of accumulation point set, and more precisely, their accumulation point sets are unbounded and indivisible continuum in complex plane. And must contain all of the escape rays as part of its set of accumulation points. The authors construct an escape ray for exponential mapping families: their set of accumulation points is a compact set in the complex plane. Furthermore, the author introduces a folding model. Three new types of escape rays are constructed. For each of these escape rays, the author defines a return sequence associated with it, according to the combined characteristics of the return sequence. The topology of the set of accumulation points is classified as follows: 1) an indivisible continuum containing partial escape rays; (2) an inseparable continuum that does not intersect with escape rays; The second part is about the dynamical system on a family of singular perturbation rational maps. The Julia set perturbed by the family of functions we constructed is the Cantor circle. But the dynamical system on the Cantor circle is not topological conjugate with the dynamic system on the Cantor circle obtained by the traditional McMullen mapping family. The authors study the iterations of this family of functions in the case of free critical point escaping to 0 or 鈭，

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