The Hausdorff Measure of the Attractor of an Iterated Function System with Parameter

Abstract: Sierpinski carpet is one of the classic fractals with strict self-similar property. In this paper, we will give a Sierpinski carpet with parameter. When parameter θ ∈ (0, pi3), the lower bound estimate for the Hausdorff measure of this set is obtained by constructing a skillful affine mapping. At the same time, the upper bound for the Hausdorff measure is estimated by the covering of k-th basic intervals. When parameter θ ∈ [pi3, pi), by a projecting mapping and the covering of k-th basic intervals,we obtain the exact value of the Hausdorff measure of the attractor of the iterated function system with parameter θ ∈ [pi3, pi)