Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

Dai, Xin-Rong; Rao, Hui; Zhang, Shuqin

doi:10.1088/1361-6544/ab1275

Cited by 5 publications

(18 citation statements)

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“…Space-filling curves (SFC) have attracted the attention of mathematicians over a century since Peano's seminal work [18]. In a series of three papers, [20], [6] and the present paper, we give a systematic investigation of space-filling curves for connected self-similar sets.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of skeleton, which can be regarded as a kind of vertex set of a fractal, was first introduced in [7], designed for SFCs of self-affine tiles. The constructions of SFCs in [20] and [6] are based on the assumption that the self-similar set in consideration possesses a skeleton.…”

Section: Introductionmentioning

confidence: 99%

“…The first author is supported by NSFC-11971195 and NSFC-11431007 and the second author is supported by project I1136 and by the doctoral program W1230 granted by the Austrian Science Fund (FWF). (c) are the approximating curves constructed by the positive Euler-tour method in [6] with this skeleton. For more details, see [20,Section 7].…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

“…(c) are the approximating curves constructed by the positive Euler-tour method in [6] with this skeleton. For more details, see [20,Section 7].…”

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confidence: 99%

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Space-Filling Curves of Self-Similar Sets (Iii): Skeletons

Rao

Zhang

2020

Fractals

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Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [6], it was shown that for all connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite type condition always possesses skeletons: an algorithm is obtained here. Introduction Space-filling curves (SFC) have attracted the attention of mathematicians over a century sincePeano's seminal work [18]. In a series of three papers, [20], [6] and the present paper, we give a systematic investigation of space-filling curves for connected self-similar sets.The notion of skeleton, which can be regarded as a kind of vertex set of a fractal, was first introduced in [7], designed for SFCs of self-affine tiles. The constructions of SFCs in [20] and [6] are based on the assumption that the self-similar set in consideration possesses a skeleton. Precisely, it is shown that Theorem 1.1 ([6]). Let K be a connected self-similar set which has a skeleton and satisfies the open set condition, then K admits space-filling curves.The goal of the present paper is to study when a self-similar set has skeletons and how to find them, which is the last part of our theory on constructing SFCs.Recall that a self-similar set is a non-empty compact set K satisfying the set equationwhere S 1 , . . . , S N are contraction similitudes on R d . The family {S 1 , . . . , S N } is called an iterated function system, or IFS in short; K is also called the invariant set of the IFS. (See for instance, [11, 9]). For the open set condition, we refer to [9, Section 9.2].

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%