本文選題：分形投影 + Hausdorff維數(shù)�。� 參考：《華中科技大學(xué)》2015年碩士論文

【摘要】：本文介紹了投影的相關(guān)理論在分形幾何中的發(fā)展,行文主要圍繞John Marstrand 1954年發(fā)表的一篇涉及到分形投影的論文展開(kāi)。本文正文主要內(nèi)容可分為3大部分。第一部分:主要介紹了Marstrand論文中的兩個(gè)著名的投影定理。第二部分:主要介紹了Kaufman給出的投影定理的一個(gè)較Marstrand給出的證明而言更具啟發(fā)性且過(guò)程更簡(jiǎn)潔的一個(gè)證明,它利用勢(shì)能定理和傅里葉變換證明了投影定理。第三部分:主要介紹了投影在發(fā)展的過(guò)程中拓展出來(lái)的一系列在集合中的理論。有整數(shù)維集合、自相似集和自仿集、隨機(jī)集中投影的一系列理論。接下來(lái)又介紹了在受限制方向上的投影理論、投影在填充維數(shù)下的相關(guān)理論、投影理論的進(jìn)一步推廣和應(yīng)用。本文首先介紹了John Marstrand的論文中介紹的關(guān)于s-集的兩個(gè)投影定理。在此基礎(chǔ)上,Roy Davies根據(jù)下面的一個(gè)結(jié)論,即具有無(wú)窮大的s-維Hausdorff測(cè)度的每一個(gè)Borel集都包含一個(gè)s-集,給出了2R中的Borel集投影到方向?yàn)?的直線(xiàn)上后其維數(shù)和測(cè)度的變化情況的一個(gè)定理。Marstrand的投影定理盡管很有用,但是由于他給出的證明過(guò)于復(fù)雜,非常不便于定理的總結(jié)和延伸,因此Kaufman利用勢(shì)能理論和傅里葉變換給出了它的一個(gè)更為簡(jiǎn)潔的證明。而Mattila從Kaufman的證明過(guò)程中獲得啟示,得到了從高維空間到子空間的投影的一些理論。另外,從Kaufman的證明過(guò)程中發(fā)現(xiàn)在某些例外集上不滿(mǎn)足投影定理,得到一系列的結(jié)論,且其中關(guān)于參數(shù)?的積分的性質(zhì)在許多其他參數(shù)化的映射中也成立,從而引出了廣義投影的概念。Jarvenpaa將投影定理的相關(guān)結(jié)論從Hausdorff維數(shù)遷移到了填充維數(shù)的條件下,得到填充維數(shù)下投影的一些結(jié)論。隨后相繼出現(xiàn)了整數(shù)維集合、自相似集和自仿集以及隨機(jī)集在受限制方向上的投影的一系列結(jié)論。
[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.
【學(xué)位授予單位】：華中科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O189

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