The purpose of this thesis is to study the concept of measure and associated concepts. The study is general in nature; that is, no particular examples of a measure are given.

Wu, Dan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 62-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Calibrated Geometry --- p.9 / Chapter 2.1 --- Theory of calibrations --- p.9 / Chapter 2.2 --- Two classical examples --- p.14 / Chapter 2.3 --- Calibrations and the VCP-forms --- p.19 / Chapter 3 --- Constructing Calibrations --- p.23 / Chapter 3.1 --- Clifford algebra and Spin groups --- p.23 / Chapter 3.2 --- Calibrations and spinors --- p.30 / Chapter 4 --- Calibrations in Exceptional Geometry --- p.39 / Chapter 4.1 --- G2 calibration --- p.40 / Chapter 4.2 --- Cayley calibration --- p.49 / Bibliography --- p.62

在該論文中，我們探討了自相似集和自仿集兩類基本分形集的拓撲結構。我們主要研究了他們的連通性、全不連通性以及李普希茲等價性。 / 我們首先研究了一類由正方塊迭代生成的自相似集的拓撲，我們稱這種自相似集為分形方塊。通過研究它的 torus-like 結構， 我們用連通分支把分形方塊的拓撲結構分成三種情形，同時我們還給出了一系列簡單有效的判別方法。這對於進一步研究其李普希茲等價類非常有用。 / 另外一個方面，基於之前對由相鄰共線性數字集生成的自仿集的連通性的研究工作，我們嘗試研究非相鄰共線性數字集，我們處理里一類特殊的由行列式絶對值為 3 的擴張矩陣 A 生成的二維自仿集。通過逐項檢驗 A 的每個特徵多項式，我們得到了該自仿集的連通性的一個完整的刻畫。 / 最近，關於自相似集的李普希茲等價的研究，引起了很多的關注。在這篇論文中，我們藉助符號空間上自帶的“擴張樹“的結構和它的雙曲邊界，拓寬了該問題的研究框架。通過邊的重排技巧，我們構造了兩個擴張樹之間的擬等距同構以便於證明他們雙曲邊界之間的李普希茲等價。最終，我們解決了更加一般的自相似集，甚至自仿集之間的李普希茲等價問題。 / In the thesis, we explore the topological structure of the self-similar sets and self-a±ne sets, two basic classes of fractals. We study their connectedness, total disconnectedness and Lipschitz equivalence. / We first initiate a new study on the topology of a class of self-similar sets generated by nested squares, which we call fractal squares. By studying the torus-like structure, we obtain some useful and simple criteria to classify the fractal squares into three types through connected components. That is very useful for the Lipschitz classification. / In another direction, motivated by the previous work on the self-affine sets associated with consecutive collinear digit sets, we make a first attempt to study the non-consecutive collinear digit sets. We deal with the special case that the planar self-a±ne sets are generated by expanding matrices A with j det(A)j = 3. By checking the characteristic polynomial of A case by case, we obtain a complete characterization for the self-a±ne sets to be connected or disconnected. / Recently there is a lot of interest to study the Lipschitz equivalence of self- similar sets. In this thesis, we provide a broader framework of the study through the concept of augmented (rooted) tree and its hyperbolic boundary. Making use of a technique of rearrangement of the edges, we construct a near-isometry between the two trees to show their boundaries are Lipschitz equivalent. Finally we establish the Lipschitz equivalence on more general self-similar sets and even self-affine sets. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Luo, Jun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 107-111). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.3 / Chapter 2 --- Topological structure of fractal squares --- p.9 / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.2 --- Classification of F by connected components --- p.11 / Chapter 2.3 --- F containing line segments --- p.17 / Chapter 2.4 --- H[superscript c] and its components --- p.19 / Chapter 2.5 --- Algorithm and Examples --- p.26 / Chapter 2.6 --- Remarks and open questions --- p.32 / Chapter 3 --- Connectedness of self-a±ne sets --- p.33 / Chapter 3.1 --- Introduction --- p.33 / Chapter 3.2 --- Preliminaries --- p.37 / Chapter 3.3 --- Integer collinear digit sets --- p.41 / Chapter 3.4 --- General collinear digit sets --- p.48 / Chapter 3.5 --- Other results on --- p.49 / Chapter 3.6 --- Remarks and open questions --- p.52 / Chapter 4 --- Lipschitz equivalence and total disconnectedness --- p.53 / Chapter 4.1 --- Introduction --- p.53 / Chapter 4.2 --- Augmented trees --- p.57 / Chapter 4.3 --- Statements of main theorems --- p.63 / Chapter 4.4 --- Proofs of main theorems --- p.72 / Chapter 4.5 --- Examples --- p.80 / Chapter 4.6 --- Remarks and open questions --- p.91 / Chapter 5 --- More on Lipschitz equivalence --- p.94 / Chapter 5.1 --- Dust-like self-similar sets --- p.94 / Chapter 5.2 --- Lipschitz classification of fractal squares --- p.99 / Bibliography --- p.107

Du, Yangge. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 88-91). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

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Bibliography : leaves 115-119

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Let (GAMMA) be a discrete group of hyperbolic isometries having a fundamental polygon with finitely many sides and no cusps. We show that the associated horocycle flow admits just one ergodic measure which is not concentrated on a single orbit. We use ergodicity of this measure to calculate the ratio set of the Patterson measure of (GAMMA).

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