The Hausdorff Dimension And Measure Of Self-Similar Sets And The Computer Realization

Fractal geometry, which was created by B.B.Mandelbrot in the 1980s, provides the idea, methods and techniques for the study of some irregular sets. As the irregular sets provide a better description of the natural phenomena than classical geometry, in recent years, fractal geometry, this rising branch of science, has gained great success while it is applied to solve a large quantity of problems of irregular geometry objects in disciplines such as mathematics, physics, chemistry, biology, engineering technology and so on. Also, the applications of fractal geometry in these areas in turn are a fruitful source of further development of it. Creation and development of fractals geometry is extremely important for the development of science.As is known, Hausdorff measure and dimension are the theory basis of fractal geometry. And the theory study on Hausdorff measure and dimensions has great importance theory and practical value. It is rather difficult to calculate and estimate the Hausdorff measure and dimension of the fractal sets. Up to now, the class of fractal sets being studied most is the self-similar set which satisfies the open set condition, and the calculation and estimation of its Hausdorff dimension has established formulas already. When it comes to measure, only a few of specific self-similar sets with less than 1 dimension has been determined. But for self-similar sets with more than 1 dimension, so far, we only estimate the upper bound or the lower bound of the Hausdorff measures of some.This thesis consists of four sections. The first part -- the introduction gives a short overview of the research current and thesignificance of fractal geometry, then it narrates this thesis's main research content and results. The second part is to give the basic concepts and theories used in this article. The concept and natures of Hausdorff measure and dimension are given first. Then it introduces the self-similar compression system, self-similar sets and open set condition. And an equivalent definition of Hausdorff dimension for self-similar set is introduced which meets the open sets condition.The third section is to study some self-similar sets on the line which meet the open set condition. First of all, the construction, measure and dimension of the classic middle third Cantor set are introduced for comparison. And then, it construct a few generalized Cantor sets and conducts preliminary research, and gives the self-similar compression system that they meet and the formulas the dimensions meet.After that it researches a special kind of generalized Cantor set in depth and gets its dimensions and measures.Some self-similar sets defined in the plane, which meet the open set condition, are reserched in the fourth section. It constructs two classes of generalized pentagonal carpets, gives the self-similar compression systems that they meet and the formulas that the dimensions meet. Then it constructs two classes of generalized hexagonal carpets, and does similar research with them.Then an in-depth research is made on a special kind of generalized snow carpet—snow carpet. The dimension has been calculated, and the upper bound of Hausdorff measure has been estimated. And it discusses how to use the computer for its upper bound estimation in the end.