Abstract
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 = ∪ n i=1 (E b,d /d + b i ) with b = (b 1,…, b n ) ∈ ℚ n 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.
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Wang, Q., Xi, L. & Zhang, K. Self-similar fractals: An algorithmic point of view. Sci. China Math. 57, 755–766 (2014). https://doi.org/10.1007/s11425-013-4767-x
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DOI: https://doi.org/10.1007/s11425-013-4767-x