Volume 30, Issue 2
A Local Property of Hausdorff Centered Measure of Self-Similar Sets

Anal. Theory Appl., 30 (2014), pp. 164-172

Published online: 2014-06

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• Abstract

We analyze the local behavior of the Hausdorff centered measurefor self-similar sets. If $E$ is a self-similar set satisfying theopen 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 thes-dimensional Hausdorff centered measure. The above inequality isused to obtain the upper bound of the Hausdorff centered measure.As the applications of above inequality, We obtained the upperbound of the Hausdorff centered measure for some self-similar setswith Hausdorff dimension equal to 1, and prove that the upperbound reach the exact Hausdorff centered measure.

• Keywords

Hausdorff centered measure Hausdorff measure self-similar sets

• AMS Subject Headings

28A78 28A80

• Copyright

COPYRIGHT: © Global Science Press

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