分形投影的发展

发布时间：2018-06-04 03:40

本文选题：分形投影 + Hausdorff维数　；参考：《华中科技大学》2015年硕士论文

【摘要】：本文介绍了投影的相关理论在分形几何中的发展,行文主要围绕John Marstrand 1954年发表的一篇涉及到分形投影的论文展开。本文正文主要内容可分为3大部分。第一部分:主要介绍了Marstrand论文中的两个著名的投影定理。第二部分:主要介绍了Kaufman给出的投影定理的一个较Marstrand给出的证明而言更具启发性且过程更简洁的一个证明,它利用势能定理和傅里叶变换证明了投影定理。第三部分:主要介绍了投影在发展的过程中拓展出来的一系列在集合中的理论。有整数维集合、自相似集和自仿集、随机集中投影的一系列理论。接下来又介绍了在受限制方向上的投影理论、投影在填充维数下的相关理论、投影理论的进一步推广和应用。本文首先介绍了John Marstrand的论文中介绍的关于s-集的两个投影定理。在此基础上,Roy Davies根据下面的一个结论,即具有无穷大的s-维Hausdorff测度的每一个Borel集都包含一个s-集,给出了2R中的Borel集投影到方向为?的直线上后其维数和测度的变化情况的一个定理。Marstrand的投影定理尽管很有用,但是由于他给出的证明过于复杂,非常不便于定理的总结和延伸,因此Kaufman利用势能理论和傅里叶变换给出了它的一个更为简洁的证明。而Mattila从Kaufman的证明过程中获得启示,得到了从高维空间到子空间的投影的一些理论。另外,从Kaufman的证明过程中发现在某些例外集上不满足投影定理,得到一系列的结论,且其中关于参数?的积分的性质在许多其他参数化的映射中也成立,从而引出了广义投影的概念。Jarvenpaa将投影定理的相关结论从Hausdorff维数迁移到了填充维数的条件下,得到填充维数下投影的一些结论。随后相继出现了整数维集合、自相似集和自仿集以及随机集在受限制方向上的投影的一系列结论。
[Abstract]:In this paper, the development of the theory of projection in fractal geometry is introduced, which is mainly focused on a paper about fractal projection published by John Marstrand in 1954. The main content of this paper can be divided into three parts. The first part mainly introduces two famous projection theorems in Marstrand's paper. The second part mainly introduces a more illuminating and concise proof of projection theorem given by Kaufman than that given by Marstrand. It proves the projection theorem by using potential energy theorem and Fourier transform. The third part mainly introduces a series of theories in the set which are developed by projection in the process of development. A series of theories for projecting integer-dimensional sets, self-similar sets, self-parody sets and random sets. Then we introduce the projection theory in the restricted direction, the theory of projection in the filling dimension, and the further extension and application of the projection theory. In this paper, we first introduce two projection theorems about s-sets, which are introduced in John Marstrand's paper. On the basis of this, we conclude that every Borel set with infinite S-dimensional Hausdorff measure contains a s-set. It is shown that the Borel set in 2R is projected to the direction of? Marstrand's projection theorem, although useful, because the proof given by Marstrand is too complex to be easily summed up and extended. Therefore, Kaufman gives a more concise proof of it by using the theory of potential energy and Fourier transform. However, Mattila gets some enlightenment from the proof of Kaufman, and gets some theories of projection from high-dimensional space to subspace. In addition, from the proof of Kaufman, we find that the projection theorem is not satisfied on some exception sets, and get a series of conclusions, among which the parameters? The concept of generalized projection. Jarvenpaa transferred the results of projection theorem from Hausdorff dimension to filling dimension, and obtained some results of projection under filling dimension. Subsequently, a series of conclusions on the projection of integer dimensional sets, self-similar sets and self-parody sets and random sets in restricted directions are presented.
【学位授予单位】：华中科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O189

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