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.

In the first chapter; we consider the invariant measure mu generated by an integral self-affine IFS. We prove that any integral self-affine measure with a common contracting matrix can be expressed as a vector-valued self-affine measure with an IFS satisfying the open set condition (OSC). The same idea can also be applied to scaling functions of refinement equations, we extended a well known necessary and sufficient condition for the existence of L1-solutions of lattice refinement equations. We then apply this vector-valued form to study the integral self-affine sets, we obtain an algorithm for the Lebesgue measure of integral self-affine region and an algorithm for the Hausdorff dimension of a class of self-affine sets. The vector-value setup also provides an easy way to consider the L q-spectrum and the multifractal formulism for self-similar measures. As an application we can conclude the differentiability of the Lq spectrum (for q > 0) of any integral self-similar measure with a common contracting matrix. / In this thesis, we study the invariant measures and sets generated by iterated function systems (IFS). The systems have been extensively studied in the frame work of Hutchinson [Hut]. For the iteration, it is often assumed that the IFS satisfies the open set condition (OSC), a non-overlap condition in the iteration. One of the advantage of the OSC is that the point in K can be uniquely represented in a symbolic space except for a mu-zero set and many important results have been obtained. Our special interest in this thesis is to transform an invariant measure with overlaps to a vector-valued form with non overlaps. The advantage of this vector-valued form is that locally the measure can be expressed as a product of matrices. / The problem considered in the third chapter is on the choice of the invariant open set in the finite type condition (FTC). From definition, the FTC depends on the choice of the invariant open set. We show that, in one dimensional case, if the IFS satisfies the FTC for some invariant open interval then it satisfies the FTC with all invariant open sets. To our surprising, we find a counter-example to show that, in high dimensional case, the invariant open set can not be chosen arbitrarily even if the IFS satisfies the OSC and generates a tile. / The second chapter is devoted to the absolute continuity of self-affine (real-valued or vector-valued) measures and some properties of the boundary of the invariant set. For self-similar IFS with a common contracting ratio, there is a necessary and sufficient condition for the self-similar measure to be absolutely continuous with respect to the Lebesgue measure (under the weak separation condition (WSC)). In our consideration we first extend the definition of WSC to self-affine IFS. Then we generalize the previous condition to obtain a necessary and sufficient condition for the self-affine vector-valued measures to be absolutely continuous with respect to the Lebesgue measure. As an application, we prove that the boundary of all integral self-affine set has zero Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding invariant (real-valued or vector-valued) measure is supported either in V or in ∂ V. / by Deng Qirong. / "March 2005." / Adviser: Ka-Sing Lau. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0301. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 87-91). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

Wong, Ting Kam Leonard. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 91-97) and index. / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Problems of fractal analysis --- p.6 / Chapter 1.2 --- The boundary theory approach --- p.7 / Chapter 1.3 --- Summary of the thesis --- p.9 / Chapter 2 --- Martin boundary --- p.13 / Chapter 2.1 --- Markov chains and discrete potential theory --- p.13 / Chapter 2.2 --- Martin compactification --- p.18 / Chapter 2.3 --- Convergence to boundary and integral representations --- p.20 / Chapter 2.4 --- Dirichlet problem at infinity --- p.25 / Chapter 3 --- Hyperbolic boundary --- p.27 / Chapter 3.1 --- Random walks on infinite graphs --- p.27 / Chapter 3.2 --- Hyperbolic compactification --- p.31 / Chapter 3.3 --- Ancona's theorem --- p.33 / Chapter 3.4 --- Self-similar sets as hyperbolic boundaries --- p.34 / Chapter 3.5 --- Hyperbolic compactification of augmented rooted trees --- p.44 / Chapter 4 --- Simple random walk on Sierpinski graphs --- p.47 / Chapter 4.1 --- Hcuristic argument for d = 1 --- p.48 / Chapter 4.2 --- Symmetries and group invariance --- p.51 / Chapter 4.3 --- Reflection principle --- p.54 / Chapter 4.4 --- Self-similar identity and hitting distribution --- p.60 / Chapter 4.5 --- Remarks and Open Questions --- p.64 / Chapter 5 --- Induced Dirichlet forms on self-similar sets --- p.66 / Chapter 5.1 --- Basics of Dirichlet forms --- p.67 / Chapter 5.2 --- Motivation: the classical Douglas integral --- p.68 / Chapter 5.3 --- Graph energy and the induced forms --- p.69 / Chapter 5.4 --- Induced Dirichlet forms on self-similar sets --- p.74 / Chapter 5.5 --- A uniform tail estimate via coupling --- p.83 / Chapter 5.6 --- Remarks and open questions --- p.89 / Index of selected terms --- p.98

A modified Witten-Sander algorithm was devised for the diffusion-limited aggregation process. The simulation and analysis were performed on a personal computer. The fractal dimension was determined by using various forms of a two-point density correlation function and by the radius of gyration. The results of computing the correlation function with square and circular windows were analyzed. The correlation function was further modified to include the edge from analysis and those results were compared to the fractal dimensions obtained from the whole aggregate. The fractal dimensions of 1.67 ± .01 and 1.75 ± .08 agree with the accepted values. Animation of the aggregation process elucidated the limited penetration into the interior and the zone of most active deposition at the exterior of the aggregate.

Thesis (M.S.)--West Virginia University, 2007. / Title from document title page. Document formatted into pages; contains x, 102 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 66-67).

Bibliography : leaves 115-119

The goal of this research project is to determine the fractal nature, if any, which surface water waves exhibit when viewed on a microscopic scale. Due to the relatively recent development of this area of mathematics, a brief introduction to the study of fractal geometry, as well as several examples of fractals, are included in this paper. From that point, this paper addresses the specific situation of a surface wave as it nears the breaking point and attempts to detect the fractal structure of a wave at this given point when viewed on a microscopic scale. This is done from both a physical standpoint based on observations at the Hinsdale Wave Facility at Oregon State University and at Cape Perpetua, Oregon on the Pacific Coast, and from a theoretical standpoint based on a spring model. / Graduation date: 1992

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