Theory Of Hausdorff Measure And Density For Fractals And Applications

Fractal theory is founded in mid-seventies and based on the Hausdorff dimensions and measures. Hausdorff dimension and measure are two basic concept of fractal geometry, also the important theory of non-linear science. Although the calculation of Hausdorff dimension made many meaningful results, but the computation of Hausdorff measure is slowly. It is extremely difficult. The present study is self-similar sets which is satisfied the conditions of open sets. The computation of Hausdorff dimension has perfect solution, but the Hausdorff measure about them is very difficult. The difficult problem is not a simple calculation, but to know enough fractal deeply, on the other word, the difficulty comes from basic theory of mathematics. We usually say that the self-similar sets have "infinite nested similar structure". Without this, what we know? Even problems for self-similar sets are not put out. The self-similar sets like the black hole which the light is not yet disclosed. The self-similar sets even like a virgin soil. Based on that, we put forward the structure of the self-similar sets, For example, some of the microstructure. The concept of upper convex density provides us with the theory and new tools which study this problem. It is closely for the calculation for The Hausdorff measures and upper convex density, in a certain sense, both equivalents. Despite this, the computation of them is very complex and difficult. Similar compression fixed point is proposed on the fractal theory which provided the theory to research the structure and properties of the fractal sets.This paper is divided into four chapters.In chapter 1, we present the introduction including the purpose and the significance of the subjects, the main content and results.In chapter 2, we construct two self-similar fractal sets which are the generalized-Sierpinski carpet and the Cartesian product of the Cantor set. Because they are satisfied the condition of open sets, so the Hausdorff dimension of them are easy calculated. According to the Part Estimation Principle and other theories, we obtain the better upper bounds of Hausdorff measure for the generalized-Sierpinski carpet and the exact value of Hausdorff measure for the Cartesian product of the Cantor set.In chapter 3, in order to further research the structure and properties of self-similar fractal sets, we introduce the concept of contracting-similarity fixed point for the self-similar sets and get some contracting -similarity fixed points for the classical fractals. In addition, we guess the whole contracting -similarity fixed points for the some self-similar sets.In chapter 4, It is closely for the calculation for The Hausdorff measures and upper convex density, in a certain sense, both calculated equivalent, so we introduce the concept of upper convex densities for the self-similar sets, and get the better estimation of lower bounds of the upper convex densities for the generalized-Sierpinski carpet and other self-similar sets.