Hausdorff Measures Of Homogeneous Perfect Sets And Hausdorff Centered Measures With Respect To Equivalent Metrics

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 continuous nor smooth 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 has appeared, such as the Haus-dorff measure, the Packing measure and the Boxing measure. However, the estimation and calculation of fractal sets is very difficult. So far, the accurate value of the Hausdorff measure of some special fractal sets have been obtained, such as the homogeneous Cantor set, some Sierpinski carpets. In this paper we provide an explicit formula for the Hausdorff measures of a class of homogeneous perfect sets. Some earlier results regarding the Hausdorff measures of Cantor-type sets are shown to be special cases of our main theorem.There are four chapters in this paper.First chapter is the introduction.In our second chapter, we first present the definitions and properties about the Hausdorff measure and Hausdorff dimension and some skills that are often used in calculating Hausdorff measures and dimensions. Also we give the definitions of self-similar sets and Moran sets and some relevant results of Hausdorff measures.Chapter 3 first gives the definition of a class of homogeneous perfect sets, then gives the exact formula of Hausdorff measures of these sets under some condition.In chapter 4, we will discuss the relations among the doubling condition, equivalence of the Hausdorff centered measures and equivalence of metrics. We will show that C^ρ,g and C^ρ,h are equivalent for any compact metric space (X,ρ) if and only if g and h are equivalent gauge functions.