Hausdorff Measure Of Two Classes Of Sierpinski Carpets

This thesis contributes to the development of calculating the Hausdorff measure of some fractal sets which are constructed by Iterated function systems, in particular the classes of Sierpinski carpet. It is easy to determine the upper bound of Hausdorff measure of these sets. In this work, we have the unite square Q=[0,1]2 and we apply two Iterated function systems to get two self-similar sets (Fractals) and we try to determine the Hausdorff measure of these sets. Firstly we show that the Hausdorff measure constructed by Method I and the Hausdorff measure constructed by Method II of these sets are equal, then we define a distribution function μ on the self-similar set constructed in R2 by an Iterated function system. Finally, we apply the mass distribution principle to this distribution function to determine the lower bound of Hausdorff measure of these sets. The first set C is a self-similar set with Hausdorff dimension s=log43 constructed from the unit square Q by an Iterated function system which consists of three contracting similarities defined as follow:The second set T is a self-similar set with Hausdorff dimension s=1 constructed from the unit square Q by an Iterated function system which consists of four contracting similarities defined as follow: In the result we get the exact Hausdorff measure of the set C, and we estimate the Hausdorff measure of the set T,...