The Hausdorff Measure Of Two Kinds Of Fractal Sets

Fractal geometry, which was created by B.B.Mandelbrot in the 1980s, provides the methods and techniques for the study of some irregular sets. As a large number of irregular sets occur in a wide variety of scientific fields and at the same time, the irregular sets provide a better description of the natural phenomena, in recent years, fractal geometry, this rising branch of mathematics, has gained great success in disciplines such as physics,chemistry,biology and so on. Also, the applications of fractal geometry in these areas in turn are a fruitful source of further development of it.This paper discusses the problem on calculation of the Hausdorff measure of two kinds of fractal sets on plane .Firstly, the Hausdorff measure of general"square flower"with various similar ratio is studied systemically; when similar ratio is in the(0,1/4], we obtain a better estimation of lower boundary of the Hausdorff measure through the theory of quality distributing and similar set OSC of general"square flower", and we obtain an estimation of upper boundary of the Hausdorff measure through constructing special cover and using the Hausdorff measure property. Meanwhile, when similar ratio is 1/4, we give an advanced estimation of upper and lower boundary of the Hausdorff measure. On the other hand, we discuss the Hausdorff measure of"cantor dust"with similar ratio1/4; Its Hausdorff measure value is found using Hausdorff measure of Sierpinski carpet and geometric similarity. At the same time, basing Lipschitz invariable property, we have a better value of its lower boundary, and through setting up an estimating formula we also have a better value of its upper boundary.This paper contains five chapters. Our first chapter introduces the research backgrou- nd of this paper. In the second chapter introduces some defines and basic theorems of the Hausdorff measure and the Hausdorff dimension. In the third chapter, we introduce self-similar sets which satisfy the open set condition some property. In the fourth chapter, we discuss the Hausdorff measure of general"square flower", and see the conclusion of table one. At last chapter, we discuss the Hausdorff measure of"cantor dust".