The Hausdorff Measure Of A Class Of Sierpinski Carpets

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, engineering technology and so on. Also, the applications of fractal geometry in these areas in turn are afruitful source of further development of it.One of the important subjects of fractal geometry is the calculation and estimation of the Hausdorff measure and dimension of the fractal sets. Up to now, some techniques for estimating the Hausdorff dimension of fractal sets have been developed, for example, as far as the self-similar set which satisfies the open set condition is concerned, its Hausdorff dimension equals to its similar dimension. The accurate value of the Hausdorff measure of some self-similar sets with Hausdorff dimension no more than 1 has been obtained, such as middle-three Cantor set [2]. In [3], the authors get the Hausdorff measure of a Sierpinski carpet whose Hausdorff dimension equals to 1. But for many other self-similar sets, so far, we only get the upper bound or the lower bound of their Hausdorff measures. For instance, in [6], the upper bound for Hausdorff measure of Sierpinski gasket is proved to be0.81794... and in [7], its lower bound - is obtained. As to Koch curve, the upperbound and lower bound for its Hausdorff measure are supposed to be 2~s/4 and 1/1.9respectively, [8][9]. In this paper, we get the exact value of the Hausdorff measure of a Sierpinski Carpet, and more important, we show the fact that the Hausdorff measure of such Sierpinski Carpet can be determined by coverings which onlyconsists of basic squares.This paper contains two chapters.Our first chapter introduces some basic and important concepts in the study of fractal geometry, such as Hausdorff measure and Hausdorff dimension, Packing measure and Packing dimension, Minkowski measure and Minkowski dimension,the relations among these measures and dimensions and the relevant theories.In the second chapter, we discuss one kind of typical fractal sets: self-similar sets which satisfy the open set condition in a profound way. The accurate value of theHausdorff measure of a class of Sierpinski Carpet has been obtained: Hs \E) = 2/2, moreover, we show thatH" (e) = Hsf(e), whereHsf(e) refers to the net measuredetermined by coverings that is made up of basic squares and s = log K— is the4Hausdorff dimension of Sierpinski Carpet ￡.