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Dimension of slices through fractals with initial cubic pattern

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Abstract

In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space Rn 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 “multi-rules” 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.

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Authors and Affiliations

  1. Department of Mathematics, Ningbo University, Ningbo, Zhejiang, 315211, China

    Lifeng Xi

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Wen Wu

  3. School of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Ying Xiong

Authors
  1. Lifeng Xi
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  2. Wen Wu
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  3. Ying Xiong
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Corresponding author

Correspondence to Wen Wu.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11371329, 11471124, 11071090, 11071224, 11101159, 11401188), K.C.Wong Magna Fund in Ningbo University, the Natural Science Foundation of Zhejiang Province (Nos. LR13A010001, LY12F02011) and the Natural Science Foundation of Guangdong Province (No. S2011040005741).

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Xi, L., Wu, W. & Xiong, Y. Dimension of slices through fractals with initial cubic pattern. Chin. Ann. Math. Ser. B 38, 1145–1178 (2017). https://doi.org/10.1007/s11401-017-1029-1

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  • DOI: https://doi.org/10.1007/s11401-017-1029-1

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