The Hausdorff Measure Of A Family Of Generalized Sierpinski Carpet

Fractal geometry, a new branch mathematics, has been developed in the last two decades. There has been a fast growth in general interest in irregular sets among researchers in many scientific fields. A fractal set is regarded as a valid physical object which is useful in the understanding of many phenomena. In recent years, fractal geometry obtained an immense success in research and application in such disciplines as physics,chemistry,material,engineering and so on. So the birth and development of fractal geometry has an extremely important function to the development of the whole science.As an important parameter to describe the fractal sets, measure and dimension plays an important role in fractal geometry. Among which Hausdorff measure is the most important one .However, the estimation and calculation of the Hausdorff measure of a general fractal set is very difficult. From now and then, there are few fractal sets succeed in calculating, at the same time ,there is no a general tool to solve the problem. In this paper, I discuss the calculation of the Hausdorff measure of a generalized Sierpinski carpet.In the first chapter, I present some basic knowledge about measures, definition and properties of Hausdorff measure and Hausdorff dimension. Also I introduce some skills that are often used in calculating the Hausdorff measure and Hausdorff dimension. Chapter 2, studies the self-similar sets and open condition. Chapter 3, introduces some theories about project of fractal sets. Chapter 4, firstly, I introduce the definition of a generalized Sierpinski carpet, by the means of upper convex density, I obtain the exact value of the Hausdorff measure of a family of Sierpinski carpet.