Dimension of Slices Through Fractals with Initial Cubic Pattern

In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R^n generated from an initial cube pattern with an （n-m）-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensi...

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Bibliographic Details
Published in:Chinese annals of mathematics. Serie B 2017-09, Vol.38 (5), p.1145-1178
Main Authors: Xi, Lifeng, Wu, Wen, Xiong, Ying
Format: Article
Language:English
Subjects:
Applications of Mathematics
Euclidean geometry
Euclidean space
Fractals
Hausdorff维数
Mathematics
Mathematics and Statistics
Self-similarity
分形维数
切片
勒贝格测度
图形
模式生成
立方体
自相似分形
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Summary:In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R^n generated from an initial cube pattern with an （n-m）-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by ＂multirules＂ take the value in Marstrand＇s theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μv is absolutely continuous with respect to the Lebesgue measure L^m. When μv〈〈L^m the connection of the local dimension of μv and the box dimension of slices is given.
ISSN:0252-9599
1860-6261
DOI:10.1007/s11401-017-1029-1