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| Dimension of Slices Through Fractals withInitial Cubic Pattern |
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Citation： |
Lifeng XI,Wen WU,Ying XIONG.Dimension of Slices Through Fractals withInitial Cubic Pattern[J].Chinese Annals of Mathematics B,2017,38(5):1145~1178 |
| Page view： 2843
Net amount： 1355 |
Authors： |
Lifeng XI; Wen WU;Ying XIONG |
Foundation： |
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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| Abstract： |
In this paper, the Hausdorff dimension of the intersection of
self-similar fractals in Euclidean space $\mathbb{R}^n$ 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 ``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
$\mu_V$ is absolutely continuous with respect to the Lebesgue
measure $\mathcal{L}^m$. When $\mu_V\ll \mathcal{L}^m$, the
connection of the local dimension of $\mu_V$ and the box dimension
of slices is given. |
Keywords： |
Slice, Self-similar set, Dimension, Fractal |
Classification： |
28A80, 37C45 |
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