Hausdorff and packing dimensions of subsets of Moran fractals with prescribed mixed group frequency of their codings
Hausdorff and packing dimensions of subsets of Moran fractals with prescribed mixed group frequency of their codings
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莫兰分形子集的豪斯多夫和包装维数及其编码的规定混合组频率
DOI：
10.1007/s00010-008-2947-5
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发表时间：
2009-03
影响因子：
0.8
通讯作者：
Li, Wenxia
中科院分区：
文献类型：
--
作者：
Wen, Zhiying;Olsen, Lars;Li, Wenxia
We compute the Hausdorff dimension and the packing dimension of subsets of Moran fractals with prescribedmixed groupfrequencies. For example, ifEdenotes the set of real numbersxin [0, 1] for which the group of digits {1, 2, 3, 4} in the decimal expansion ofxoccurs with relative frequencyand the group of digits {0, 1, 2, 8, 9} in the decimal expansion ofxoccurs with relative frequency, then our results shows that $$ {\rm dim_{\sf H}} E = {\rm dim_{\sf P}} E = - {\frac {1} {log 10}} log \left( {\frac {t^{t_1}_{1} t^{t_2}_{2} (1 - t_1)^{1-t_1} (1 - t_2)^{1-t_2}} {2^{t_1}3^{1-t_1}}}\right) $$, where dimHdenotes the Hausdorff dimension and dimPdenotes the packing dimension. Observe that the two groups of digits with prescribed frequencies, namely {1, 2, 3, 4} and {0, 1, 2, 8, 9}, aremixed, i.e. they are not disjoint. Previous work [LD, O1, V] has investigated the non-mixed case. In this paper we investigate the more difficult problem of finding the Hausdorff dimension and packing dimension of subsets of Moran fractals with prescribedmixed groupfrequencies.
We compute the Hausdorff dimension and the packing dimension of subsets of Moran fractals with prescribedmixed groupfrequencies. For example, ifEdenotes the set of real numbersxin [0, 1] for which the group of digits {1, 2, 3, 4} in the decimal expansion ofxoccurs with relative frequencyand the group of digits {0, 1, 2, 8, 9} in the decimal expansion ofxoccurs with relative frequency, then our results shows that $$ {\rm dim_{\sf H}} E = {\rm dim_{\sf P}} E = - {\frac {1} {log 10}} log \left( {\frac {t^{t_1}_{1} t^{t_2}_{2} (1 - t_1)^{1-t_1} (1 - t_2)^{1-t_2}} {2^{t_1}3^{1-t_1}}}\right) $$, where dimHdenotes the Hausdorff dimension and dimPdenotes the packing dimension. Observe that the two groups of digits with prescribed frequencies, namely {1, 2, 3, 4} and {0, 1, 2, 8, 9}, aremixed, i.e. they are not disjoint. Previous work [LD, O1, V] has investigated the non-mixed case. In this paper we investigate the more difficult problem of finding the Hausdorff dimension and packing dimension of subsets of Moran fractals with prescribedmixed groupfrequencies.
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