The Centered Covering Measures of a Class of Self-Similar Sets on the Plane

Abstract

Let $S \subset R^2$ be the attractor of the iterated function system $\{f_1,f_2,f_3, f_4\}$, where $f_i(x)=\lambda_i x+b_i$, $i=1,2,3,4$, $x=(x_1,x_2) \in R^2$, $0<\lambda_i \le \frac{1}{2+\sqrt{2}}$,$b_1=(0,0)$, $b_2=(1-\lambda_2, 0)$, $b_3=(1-\lambda_3,1-\lambda_3)$, and $b_4=(0, 1-\lambda_4)$. This paper determines the exact centered covering measure of $S$ under some conditions relating to the contraction parameters.