No description available.

The purpose of this paper is to explore some of the properties of the Cantor set and to extend the idea of this set to metric spaces, in general, and to other sets of real numbers and sets in N-dimensional Euclidean space, in particular.

The objective of my thesis is to find optimal points and the quantization error for a probability measure defined on a Cantor set. The Cantor set, we have considered in this work, is generated by two self-similar contraction mappings on the real line with distinct similarity ratios. Then we have defined a nonhomogeneous probability measure, the support of which lies on the Cantor set. For such a probability measure first we have determined the n-optimal points and the nth quantization error for n = 2 and n = 3. Then by some other lemmas and propositions we have proved a theorem which gives all the n-optimal points and the nth quantization error for all positive integers n. In addition, we have given some properties of the optimal points and the quantization error for the probability measure. In the end, we have also given a list of n-optimal points and error for some positive integers n. The result in this thesis is a nonhomogeneous extension of a similar result of Graf and Luschgy in 1997. The techniques in my thesis could be extended to discretise any continuous random variable with another random variable with finite range.

Este trabalho está preocupado com o conceito de medida e dimensão de Hausdorff usando ferramentas matemáticas adequadas. Como, frequentemente, é importante e difícil determinar a dimensão Hausdorff 1 de um conjunto e ainda mais difícil de encontrar ou mesmo estimar a sua medida Hausdorff, por auto proteção é usado o conjunto ternário de Cantor. A construção ternária simplifica certas dificuldades técnicas sobre a teoria da dimensão. O conjunto de Cantor é um exemplo interessante de um conjunto magro, perfeito, compacto e não enumerável, cuja medida e dimensão topológica são nulas. A análise de muitas das suas propriedades e consequências interessantes nos campos da teoria dos conjuntos e da topologia nos oferece uma rota direta que leva à medida Hausdorff do conjunto Cantor e sua dimensão fractal que é igual à sua dimensão Hausdorff. Também é calculada a dimensão Hausdorff para alguns fractais clássicos, como o tapete Sierpinski e a curva de flocos de neve von Koch. / This work is concerned with the concept of Hausdorff measure and dimension using suitable mathematical tools. Since it is often important and dificult to determine the Hausdorff dimension2 of a set and even harder to find or even to estimate its Hausdorff measure, by self-protection choices, it is used the ternary Cantor set. The ternary construction reduces technical difficulties about dimension theory. Cantor set is an interesting example of a meager, perfect, compact, uncountable set whose measure and topologic dimension are zero. Analysis of many of its interesting properties and consequences in the fields of set theory and topology provides a direct route that leads to the Hausdorff measure of the Cantor set and its fractal dimension that is equal to its Hausdorff dimension. It is also computed the Hausdorff dimension for some classical fractals such as the Sierpinski carpet and the von Koch snowflake curve.

Holomorphic correspondences are multivalued functions from the Riemann sphere to itself. This thesis is concerned with a certain type of holomorphic correspondence known as a covering correspondence. In particular we are concerned with a one complexdimensional family of correspondences constructed by post-composing a covering correspondence with a conformal involution. Correspondences constructed in this manner have varied and intricate dynamics. We introduce and analyze two subfamilies of this parameter space. The first family consists of correspondences for which the limit set is a Cantor set, the second family consists of correspondences for which the limit set is connected and for which the action of the correspondence on the complement of this limit set exhibits certain group like behaviour.

An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.

vii, 124 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation is a study of the relationship between minimal dynamical systems on the product of the Cantor set ( X ) and torus ([Special characters omitted]) and their corresponding crossed product C *-algebras. For the case when the cocyles are rotations, we studied the structure of the crossed product C *-algebra A by looking at a large subalgebra A x . It is proved that, as long as the cocyles are rotations, the tracial rank of the crossed product C *-algebra is always no more than one, which then indicates that it falls into the category of classifiable C *-algebras. In order to determine whether the corresponding crossed product C *-algebras of two such minimal dynamical systems are isomorphic or not, we just need to look at the Elliott invariants of these C *-algebras. If a certain rigidity condition is satisfied, it is shown that the crossed product C *-algebra has tracial rank zero. Under this assumption, it is proved that for two such dynamical systems, if A and B are the corresponding crossed product C *-algebras, and we have an isomorphism between K i ( A ) and K i ( B ) which maps K i (C(X ×[Special characters omitted])) to K i (C( X ×[Special characters omitted])), then these two dynamical systems are approximately K -conjugate. The proof also indicates that C *-strongly flip conjugacy implies approximate K -conjugacy in this case. We also studied the case when the cocyles are Furstenberg transformations, and some results on weakly approximate conjugacy and the K -theory of corresponding crossed product C *-algebras are obtained. / Committee in charge: Huaxin Lin, Chairperson, Mathematics Daniel Dugger, Member, Mathematics; Christopher Phillips, Member, Mathematics; Arkady Vaintrob, Member, Mathematics; Li-Shan Chou, Outside Member, Human Physiology

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.

Seja K um conjunto de Cantor. Neste trabalho apresentamos dois teoremas relacionados a densidade do conjunto das dinâmicas simbólicas. No caso de endomorfismos provamos que, dado uma dinâmica T : K K, existe uma T : K K próxima a T, tal que toda órbita é finalmente periódica. Já no caso de homeomorfismos, mostramos que, dado uma dinâmica T : K K, existe uma T : K K próxima a T, tal que o w-limite de toda órbita de T é uma órbita periódica. Em particular, mostramos que, em ambos os casos, todas as medidas ergódicas estão suportadas em órbitas periódicas. / Let K be a Cantor set. In this work we present two theorems related to the density of symbolic dynamics. We prove that given an endomorphism T : K K then there exists an endomorphism ~ T : K K close to T such that every orbit is finally periodic. We also prove that given a homeomorphism T : K K then there exists a homeomorphism ~ T : K K close to T such that the w-limit of every orbit is a periodic orbit. In particular, we have shown, in both cases, that all ergodic measures have support on periodic orbits.

Seja K um conjunto de Cantor. Neste trabalho apresentamos dois teoremas relacionados a densidade do conjunto das dinâmicas simbólicas. No caso de endomorfismos provamos que, dado uma dinâmica T : K K, existe uma T : K K próxima a T, tal que toda órbita é finalmente periódica. Já no caso de homeomorfismos, mostramos que, dado uma dinâmica T : K K, existe uma T : K K próxima a T, tal que o w-limite de toda órbita de T é uma órbita periódica. Em particular, mostramos que, em ambos os casos, todas as medidas ergódicas estão suportadas em órbitas periódicas. / Let K be a Cantor set. In this work we present two theorems related to the density of symbolic dynamics. We prove that given an endomorphism T : K K then there exists an endomorphism ~ T : K K close to T such that every orbit is finally periodic. We also prove that given a homeomorphism T : K K then there exists a homeomorphism ~ T : K K close to T such that the w-limit of every orbit is a periodic orbit. In particular, we have shown, in both cases, that all ergodic measures have support on periodic orbits.