令B{U+209C}表示一個平面布朗運動。我們把C \B[0, 1] 的無界連通分支的邊界稱爲B[0; 1] 的外邊界。在本文中，我們將討論如何計算B[0,1] 的外邊界的Hausdorff 維數。 / 我們將在第二章討論Lawler早期的工作[7]。他定義了一個常數ζ(所謂的不聯通指數） 。利用能量的方法， 他證明了 B[0,1]的外邊界的Hausdorff維數是2(1 - ζ)概率大於零， 然後0-1律可以明這個概率就是1。但是用他的方法我們不能算出ζ的準確值。 / Lawler, Schramm and Werner 在一系列文章[10]，[11] 和[13] 中研究了SLE{U+2096}和excursion 測度。利用SLE6 和excursion 測度的共形不變性，他們可以計算出了布朗運動的相交指數ξ (j; λ )。因此ζ = ξ (2; 0)/2 = 1/3，由此可以知道B[0, 1] 的外邊界的Hausdorff 維數就是4/3。從而可以說完全證明了著名的Mandelbrot 猜想。 / Let B{U+209C} be a Brownian motion on the complex plane. The frontier of B[0; 1] is defined to be the boundary of the unbounded connected component of C\B[0; 1].In this thesis, we will review the calculation of the Hausdorff dimension of the frontier of B[0; 1]. / We first dissuss the earlier work of Lawler [7] in Chapter 2. He defined a constant ζ (so called the dimension of disconnection exponent). By using the energy method, he proved that with positive probability the Hausdorff dimension of the frontier of B[0; 1] is 2(1 -ζ ), then zero-one law show that the probability is one. But we can not calculate the exact value of ζ in this way. / In the series of papers by Lawler, Schramm and Werner [10], [11] and [13], they studied the SLE{U+2096} and excursion measure. By using the conformal invariance of SLE₆ and excursion measure, they can calculate the exact value of the Brownian intersection exponents ξ(j, λ). Consequently, ζ = ξ(2, 0)/2 = 1/3, and the Hausdorff dimension of the frontier of B [0,1] is 4/3 almost surely. This answers the well known conjecture by Mandelbrot positively. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Pengfei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 53-55). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Hausdorff dimension of the frontier of Brownian motion --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Hausdorff dimension of Brownian frontier --- p.13 / Chapter 3 --- Stochastic Loewner Evolution --- p.24 / Chapter 3.1 --- Definitions --- p.24 / Chapter 3.2 --- Continuity and Transience --- p.26 / Chapter 3.3 --- Locality property of SLE₆ --- p.30 / Chapter 3.4 --- Crossing exponent for SLE₆ --- p.32 / Chapter 4 --- Brownian intersection exponents --- p.37 / Chapter 4.1 --- Half-plane exponent --- p.37 / Chapter 4.2 --- Whole-plane exponent --- p.41 / Chapter 4.3 --- Proof of Theorem 4.6 and Theorem 4.7 --- p.44 / Chapter 4.4 --- Proof of Theorem 1.2 --- p.47 / Chapter A --- Excursion measure --- p.48 / Chapter A.1 --- Metric space of curves --- p.48 / Chapter A.2 --- Measures on metric space --- p.49 / Chapter A.3 --- Excursion measure on K --- p.49 / Bibliography --- p.53

Chan, Ying Ying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 66-69). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Weierstrass function --- p.7 / Chapter 1.2 --- Rademacher series --- p.10 / Chapter 2 --- Preliminaries --- p.12 / Chapter 2.1 --- Hausdorff dimension and box dimension .. --- p.12 / Chapter 2.2 --- Properties of Hausdorff dimension and box dimension --- p.15 / Chapter 2.3 --- Basic techniques in computing dimensions . --- p.16 / Chapter 2.4 --- Graphs of functions --- p.18 / Chapter 3 --- Weierstrass Function --- p.20 / Chapter 3.1 --- Weierstrass-like functions and their box dimension --- p.20 / Chapter 3.2 --- Hausdorff dimension of Weierstrass-like graphs --- p.23 / Chapter 3.3 --- Weierstrass function with a random phase angle --- p.31 / Chapter 4 --- Rademacher series --- p.37 / Chapter 4.1 --- Basic properties --- p.38 / Chapter 4.2 --- Box dimension for Rademacher series with generalization --- p.39 / Chapter 4.3 --- Some remainders on the infinite Bernoulli convolution --- p.46 / Chapter 5 --- Rademacher series with Pisot reciprocal as parameter --- p.48 / Chapter 5.1 --- Pisot number --- p.48 / Chapter 5.2 --- Hausdorff dimension --- p.49 / Chapter 5.3 --- Matrix representation --- p.54 / Chapter 5.3.1 --- Set-up --- p.54 / Chapter 5.3.2 --- Case of golden ratio --- p.61

Thesis (M.A.)--York University, 2008. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 166-167). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:MR45921

The goal of this thesis is to provide a unified framework in which to analyze the dynamics of two seemingly unrelated families of one-dimensional dynamical systems, namely the family of quadratic polynomials and continued fractions. We develop a combinatorial calculus to describe the bifurcation set of both families and prove they are isomorphic. As a corollary, we establish a series of results describing the behavior of entropy as a function of the parameter. One of the most important applications is the relation between the topological entropy of quadratic polynomials and the Hausdorff dimension of sets of external rays landing on principal veins of the Mandelbrot set. / Mathematics

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 60-61). Also available on the Internet.

Zugl.: Kaiserslautern, Techn. Univ., Diss., 2009

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 60-61). Also available on the Internet.

Th. doct.--Math. appl.--Paris 9, 1996. / Bibliogr. p. 199-204. Résumé. 1996 d'après la déclaration de dépôt légal.

Thesis (Ph.D)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Dr. J. R. Jackson; Committee Member: Dr. D. Scott; Committee Member: Dr. D. V. Anderson; Committee Member: Dr. P. Vela; Committee Member: Dr. R. Mersereau. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Dresden, Techn. Univ., Diss., 1998.