A Local Property of Hausdorff Centered Measure of Self-Similar Sets

Abstract

We analyze the local behavior of the Hausdorff centered measure for self-similar sets. If $E$ is a self-similar set satisfying the open set condition, then$$C^s(E \cap B(x,r)) \le (2r)^s$$for all $x \in E$ and $r >0$, where $C^s$ denotes the $s$-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.