On multifractal of Cantor dust |
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Authors: | In-Soo Baek |
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Affiliation: | (1) Department of Mathematics, Pusan University of Foreign Studies, Pusan, 608-738, Republic of Korea |
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Abstract: | We consider quasi-self-similar measures with respect to all real numbers on a Cantor dust. We define a local index function on the real numbers for each quasi-self-similar measure at each point in a Cantor dust. The value of the local index function at the real number zero for all the quasi-self-similar measures at each point is the weak local dimension of the point. We also define transformed measures of a quasi-self-similar measure which are closely related to the local index function. We compute the local dimensions of transformed measures of a quasi-self-similar measure to find the multifractal spectrum of the quasi-self-similar measure. Furthermore we give an essential example for the theorem of local dimension of transformed measure. In fact, our result is an ultimate generalization of that of a self-similar measure on a self-similar Cantor set. Furthermore the results also explain the recent results about weak local dimensions on a Cantor dust. This work is supported by the Korea Research Foundation Grant (KRF-2005-013-C00004) |
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Keywords: | Cantor set Hausdorff dimension packing dimension quasi-self-similar measure local dimension |
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