Gauges for the self‐similar sets |
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| Authors: | Sheng‐You Wen Zhi‐Xiong Wen Zhi‐Ying Wen |
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| Institution: | 1. Department of Mathematics, Hubei University, Hubei 430062, P. R. China;2. Phone: +86 27 88661740 3610;3. Department of Mathematics, Wuhan University, Hubei 430072, P. R. China;4. Phone: +86 27 68766400;5. Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China |
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| Abstract: | For a self‐similar set E with the open set condition we completely determine the class of its Hausdorff gauges and the class of its prepacking gauges. Moreover, its Hausdorff measures and its packing premeasures with respect to the corresponding gauges are estimated. Without the open set condition we prove that a doubling gauge function is a packing gauge of E if and only if it is a prepacking gauge of E. Also, we give some extensions and applications of these results. Here a gauge function is called a Hausdorff, a prepacking, and a packing gauge of a set, if with respect to the function the set has positive and finite Hausdorff measure, packing premeasure, and packing measure, respectively. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
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| Keywords: | Self‐similar set open set condition gauge function doubling condition |
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