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Master's Dissertation
DOI
10.11606/D.55.2019.tde-04012019-151235
Document
Author
Full name
Rita de Cássia Morasco da Cruz
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2018
Supervisor
Committee
Santos, Jair Silverio dos (President)
Mitrowsky, Rafael Andres Rosales
Nicola, Selma Helena de Jesus
Silva, Aparecida Francisco da
Title in Portuguese
Geometria Fractal: conjunto de Cantor, dimensão e medida de Hausdorff e aplicações
Keywords in Portuguese
Conjunto de cantor
Dimensão
Fractal
Medida
Abstract in Portuguese
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.
Title in English
Fractal Geometry: Cantor set, Hausdorff dimension and masurement and applications
Keywords in English
Cantor set
Dimension
Fractal
Measure
Abstract in English
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.
 
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Publishing Date
2019-01-04
 
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