Self-similar fractals： An algorithmic point of view

This paper studies the self-similar fractals with overlaps from an algorithmic point of view.A decidable problem is a question such that there is an algorithm to answer＂yes＂or＂no＂to the question for every possible input.For a classical class of self-similar sets{E b.d}b,d where E b.d=Sn i=1（E b,d/d＋...

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Bibliographic Details
Published in:Science China. Mathematics 2014-04, Vol.57 (4), p.755-766
Main Authors: Wang, Qin, Xi, LiFeng, Zhang, Kai
Format: Article
Language:English
Subjects:
Algorithms
Applications of Mathematics
Fractal analysis
Fractals
Hausdorff维数
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Self-similarity
Similarity
Texts
多项式时间算法
开集条件
强分离条件
测试
相似性
自相似分形
自相似集
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Summary:This paper studies the self-similar fractals with overlaps from an algorithmic point of view.A decidable problem is a question such that there is an algorithm to answer＂yes＂or＂no＂to the question for every possible input.For a classical class of self-similar sets{E b.d}b,d where E b.d=Sn i=1（E b,d/d＋b i）with b=（b1,...,b n）∈Qn and d∈N∩[n,∞）,we prove that the following problems on the class are decidable：To test if the Hausdorff dimension of a given self-similar set is equal to its similarity dimension,and to test if a given self-similar set satisfies the open set condition（or the strong separation condition）.In fact,based on graph algorithm,there are polynomial time algorithms for the above decidable problem.
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-013-4767-x