Year: 2014
Analysis in Theory and Applications, Vol. 30 (2014), Iss. 2 : pp. 164–172
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2014.v30.n2.3
Analysis in Theory and Applications, Vol. 30 (2014), Iss. 2 : pp. 164–172
Published online: 2014-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 9
Keywords: Hausdorff centered measure Hausdorff measure self-similar sets.