An Estimate And A Calculation About Hausdorff Measures And Upper Convex Density Of Self Similar Sets

In this thesis, we mainly study how to calculate the exact Hausdorff measure of a class of self-similar sets and how to estimate the upper bound of upper convex density of some concrete points in some self-similar sets. In addition, we present a necessary and sufficient condition for a class of self-similar sets on real line to have best coverings and some sufficient conditions for an almost everywhere best covering to be a best one for the self-similar sets satisfying the open set condition. This thesis is divided into four chapters.In the first chapter we introduce the research background about fractal, and describe a detailed basic definitions and lemmas about fractal.In the second chapter we study the problem of calculation of Hausdorff measure of a class of self-similar sets in a regular hexahedron. When the similar ratios satisfies certain conditions, we prove that the natural cover is the best shape for upper convex density. It means that the natural cover is a best one, then the exact value of Hausdorff measure of this class of self-similar sets is ((?))s, where s is Hausdorff measure.In the third chapter we first prove that the family of best shaps of the vertexs of a class of Sierpinski carpets is a covering of the Sierpinski carpets, then prove the minimum of the upper convex density is achieved at their vertexs. This result extends some recent results.In the forth chapter, we first take advantaged of an important characteristic that the best shap of self-similar sets on real line must be closed interval, then present a necessary and sufficient condition for a class of self-similar sets on real line to have best coverings, and some sufficient conditions for an almost everywhere best covering to be a best one for the self-similar sets satisfying the open set condition.