分形集的最佳参数化和space-filling曲线

发布时间：2018-06-12 22:21

本文选题：Space-filling曲线 + 自相似集　；参考：《华中师范大学》2016年博士论文

【摘要】：在本学位论文中,我们主要研究自相似集的最佳参数化,所谓最佳参数化,也就是几乎处处一对一,在一定意义下保测和1/3-Holder连续,这里S是自相似集的豪斯道夫维数.特别地,在本文中,我们构造出一套系统且严格的方法,能够对空间填充曲线有更深入的了解.通过这套方法,我们设计出了一套程序,只要输入相应的数据,就能得到对应的最佳参数化.首先,我们引进一个概念Hnear GIFS(带有序结构的图递归函数迭代系统).我们证明了，，当一个linear GIFS的不变集满足开集条件时,它们有最佳参数化!因此,构造不变集的空间填充曲线,也就是需要造出一个恰当的linear GIFS.其次,为了研究自相似集的linear GIFS结构,我们引进不变集的skeleton这个新的定义.一个给定的有限skeleton决定一个完全有向图Go.将G0代入函数迭代系中迭代,我们得到一个由G0的自仿像构成的有向图.通过研究G0和这个新的有向图的关系,我们可以得到不同的欧拉代换规则,因此可以诱导不同的linear GIFS.这里验证开基条件的问题可以由分形几何的理论解决.这样我们就证明了,在一定的假设条件下,满足开集条件和有有限skeleton的自相似集,有最佳参数化.同时,本文还对经典的空间填充曲线进行了比较详细的研究,并且给出了构造经典的空间填充曲线的三种方法.在本文的最后一部分,我们会证明前面给出的一些假设条件是多余的,并证明了在满足开集条件和有限skeleton条件的下,不变集有最佳参数化.这也说明了我们的理论囊括了自相似集中的很大一类,其中也包括满足有限型条件的自相似集,并且我们的结果几乎包含了之前在这方面的所有结论.
[Abstract]:In this dissertation, we mainly study the optimal parameterization of self-similar sets, the so-called optimal parameterization, that is, almost everywhere one-to-one, in a certain sense, preserving and 1 / 3-Holder continuity, where S is the Hausdorf dimension of the self-similar set. In particular, in this paper, we construct a systematic and strict method to understand the space filling curve more deeply. Through this method, we design a program, which can get the best parameterization by inputting the corresponding data. First, we introduce the concept Hnear GIFS (Graph Recursive function iterative system with ordered structure). We prove that when an invariant set of linear GIFS satisfies the open set condition, they have the best parameterization! Therefore, it is necessary to construct an appropriate linear GIFSs to construct the space filling curve of the invariant set. Secondly, in order to study the linear GIFS structure of self-similar sets, we introduce a new definition of skeleton of invariant sets. A given finite skeleton determines a completely directed graph Go. By substituting G0 into the iterative system of functions, we obtain a directed graph composed of the self-imitating image of G0. By studying the relation between G0 and this new digraph, we can obtain different Euler substitution rules, so we can induce different linear GIFSs. The problem of verifying the open basis condition can be solved by the theory of fractal geometry. In this way, we prove that under certain assumptions, the open set condition and the self-similar set with finite skeleton have the best parameterization. At the same time, the classical space filling curve is studied in detail, and three methods of constructing the classical space filling curve are given. In the last part of this paper, we prove that some of the assumptions given above are superfluous, and prove that the invariant set has the best parameterization under the open set condition and the finite skeleton condition. This also shows that our theory contains a large class of self-similar sets, including self-similar sets satisfying finite type conditions, and our results contain almost all previous results in this respect.
【学位授予单位】：华中师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O189

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