Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 99-103).

We introduce a theory of contractive Markov systems (CMS) which provides a unifying framework in so-called "fractal" geometry. It extends the known theory of iterated function systems (IFS) with place dependent probabilities [1][8] in a way that it also covers graph directed constructions of "fractal" sets [18]. Such systems naturally extend finite Markov chains and inherit some of their properties. In Chapter 1, we consider iterations of a Markov system and show that they preserve the essential structure of it. In Chapter 2, we show that the Markov operator defined by such a system has a unique invariant probability measure in the irreducible case and an attractive probability measure in the aperiodic case if the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero, and the system satisfies a condition of a contractiveness on average. This generalizes a result from [1]. Furthermore, we show that the rate of convergence to the stationary state is exponential in the aperiodic case with constant probabilities and a compact state space. In Chapter 3, we construct a coding map for a contractive Markov system. In Chapter 4, we calculate Kolmogorov-Sinai entropy of the generalized Markov shift. In Chapter 5, we prove an ergodic theorem for Markov chains associated with the contractive Markov systems. It generalizes the ergodic theorem of Elton [8].

The basic 'fractal percolation' process was first proposed by Mandelbrot in 1974 and takes the following form. Let M ≥2 and P ∈ [0,1]; we start with the unit square C0 = [0,1]2; Divide C0 into M2 equal closed squares, each of side-length M−1 , in the natural way and retain each of these squares with probability p, or else remove it with probability 1 - p. We let C1 be the union of those squares retained. The process is now repeated within each square of C1 to give a new set C2⊆C1, consisting of squares of side-length M−2. Iterating the construction in the obvious way, we obtain a decreasing sequence of sets C0⊇ C1 ⊇ C2 ⊇ ... with limit C[sub]∞ = ∩[sub]n≥1C[sub]n. The set C[sub]∞ is an example of a random Cantor set, and is typically highly intricate in nature. It may be empty, dust-like or highly connected, depending on the value of p; percolation is said to occur if C[sub]∞ contains large connected components linking opposite sides of the unit square. In this thesis we shall investigate some of the geometrical and topological properties of C[sub]∞ that hold either almost surely (with probability 1) or with non-zero probability. In particular, the following results are established. We obtain (almost sure) lower and upper bounds on the box-counting dimension of the 'straightest' crossings in C[sub]∞ whenever percolation occurs; we also look at the distribution of the sizes of the connected components and the probability of percolation. In the three-dimensional version of the process, we establish the existence of two distinct phases of percolation, corresponding to the occurrence of paths and surfaces (or 'sheets') in the limit set, and study the limiting behaviour of the phase transition to sheet percolation as M → ∞. We also consider the results of some computer simulations of fractal percolation and present a number of generalisations of the basic process and other closely related constructions.

In oil exploration, changes in soil depth or thickness of a rock are indicators for the presence of the so called "seismic horizons" that identify the possible presence of oil. The data concerned are obtained by making measurements at randomly distributed sparse points. It is of interest to reconstruct the full model of the surface of the rock or the terrain, from the knowledge of the few sparse data points. This reconstruction cannot be achieved by using ordinary interpolation methods, as these methods assume that the reconstructed surface is smooth. Instead, a fractal prior model for the terrain has to be assumed. A constraint fractal formation then follows, with the constraints being the data points available. The dimension of the fractal used is inferred from the data points that are available, on the basis of the assumption that a fractal model applies and from the fact that a fractal exhibits the same properties at all scales. Several tools for the creation of artificial fractals of varying degrees of roughness are used to give a wide range of data for the reconstruction experiments. A tool to measure the fractal dimension of a surface, or a set of sparse data points, is an important part of the reconstruction process. Several methods of fractal dimension measurement are developed and thoroughly tested with many different surfaces. The reliability of the dimension calculation and how this changes with different levels of sparsity is investigated. Both tools are then modified to enable the production and measurement of anisotropic fractals - fractals with different levels of roughness in different directions. These sorts of fractal surfaces have received little or no attention in the literature and fractal reconstructions using prior knowledge of the anisotropy have not been done before. Several different versions of the fractal reconstruction method are developed and the control of the dimension of the reconstructed surface is carefully investigated. Example reconstructions are then presented, using both artificial and real fractals. The subsampling of the data is performed both at random and in regular patterns and the reconstruction is forced to extrapolate from as well as interpolate between the data points. Finally the reconstruction method is modified to incorporate knowledge of any anisotropy in the fractal surface. The method is tested on both real and artificial data and shows significant advantages over the regular isotropic reconstruction.

Thesis (M.S.)--University of Southern California, 1998. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.S.)--University of Southern California, 1998. / Includes bibliographical references.

In this paper, we begin in Chapter 1 by giving a brief overview of the history of fractal geometry, focusing on six of the most important mathematicians in this field. Chapter 2 explains the main definitions needed for the remainder of the paper. In Chapter 3, we clarify the process of creating fractals from iterated function systems. Chapter 4 consists of an examination of the properties of traditional fractals. Next, in Chapter 5, we examine the newly discovered V-variable fractals and their properties. Finally we consider applications and future research in the field of fractal geometry.

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Xiao, Chang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 37-[38]). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Yung Po-lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 62-64). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The basics of doubling measures --- p.7 / Chapter 2.1 --- Existence of doubling measures --- p.7 / Chapter 2.2 --- More examples of doubling measures --- p.16 / Chapter 3 --- Doubling of self-similar measures --- p.20 / Chapter 3.1 --- Open set condition and doubling --- p.24 / Chapter 3.2 --- Examples of doubling with OSC --- p.31 / Chapter 3.3 --- Bernoulli convolution and golden ratio --- p.41 / Chapter 4 --- Applications of doubling measures --- p.48 / Chapter 4.1 --- Singular integral operators --- p.48 / Chapter 4.2 --- Poincare inequalities and local Sobolev embedding --- p.56 / Chapter 4.3 --- Remarks --- p.60 / Bibliography --- p.62