Hausdorff Measures Of Two Classes Of Fractal Sets

Fractal geometry, a new branch of mathematics, has been developed in the last two decades. There has been a fast growth in general interest in irregular sets which are neither smooth nor surfacelike among researchers in many scientific fields. A fractal set is regarded as a valid physical object which is useful in the understanding of many scientific phenomena, such as the Brownian motion of particles, turbulence in fluids, the growth of plants, geographical coastlines and surfaces. In recent years, fractal geometry obtained an immense success in research and application in such disciplines as mathematics, physics, chemistry, biology, medical science, geology, material, engineering and so on. At the same time, a large number of questions that are put forward in different disciplines stimulate the thorough development of fractal geometry. So the birth and development of fractal geometry have an extremely important function to the development of the whole science.As an important parameter to describe the fractal sets, measure plays an important role in fractal geometry. Nowadays, measure of different forms hasappeared, such as the Hausdorff measure, the Packing measure and the Minkowski measure, among which the Hausdorff measure is the most important one.However, the estimation and calculation of the Hausdorff measure of fractal sets is very difficult. So far, the accurate value of the Hausdorff measure of some special fractal sets with Hausdorff dimension no more than 1 has been obtained, such as the homogeneous Cantor set, some Sierpinski carpets. In this paper, we obtain the exact Hausdorff measure of the homogeneous Cantor set and a class of Moran sets. There are three chapters in this paper.In our first chapter, we present some basic knowledge about measures, the definitions and properties about the Hausdorff measure and Hausdorff dimension. Also we introduce some skills that are often used in calculating the Hausdorff measure and Hausdorff dimension.Chapter 2 studies the Hausdorff measure of the homogeneous Cantor set. Firstly, the relevant results about the calculation of the Hausdorff measure of self-similar sets are introduced. Then the lower bound for Hausdorff measures of the homogeneous Cantor sets is given by an elementary method. At the same time , the upper bound for their Hausdorff measures is obtained by the covering of kth basic intervals, and finally the exact Hausdorff measures of these sets is obtained.In chaper 3, firstly, we introduce the definition of Moran set and the Hausdorff dimension of some Moran sets. Then we obtain the exact value of the Hausdorff measure of a class of Moran set by the same means as calculating the homogeneous Cantor set in chapter 2.