Space-filling curves of self-similar sets (III): Skeletons

Rao, Hui; Zhang, Shuqin

doi:10.48550/arxiv.1804.10480

Cited by 1 publication

(4 citation statements)

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“…Self-similar sets of finite type is an important class of fractals, see for instance [8,27,36]. In a subsequent paper [38], we show that if K is a self-similar set of finite type, then K possesses skeletons. Consequently, Theorem 1.4.…”

Section: A Natural Selection Of the Initial Graph Ismentioning

confidence: 79%

“…In this section, we give a brief introduction to skeletons of self-similar sets. A detailed study is carried out in Rao and Zhang [38].…”

Section: Skeleton Of Self-similar Setmentioning

confidence: 99%

“…Using neighbour graph of self-similar sets, it is shown that a connected self-similar set of finite type always possesses skeletons ( [38]), and actually an algorithm of finding skeletons is given there. There do exist self-similar sets without skeletons (see figure 8).…”

Section: Proposition 23 ([38]mentioning

confidence: 99%

“…Space-filling curves (SFC) have fascinated mathematicians for more than a century since Peano's monumental work in 1890. In a series of three papers ( [37], the present paper, and [38]), we develop a theory to construct SFCs of self-similar sets. For a given connected selfsimilar set, finding substitution rules leading to SFCs is a long-standing and difficult problem, and it is the main concern of the present paper.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper.Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the positive Euler-tour class. Finally, we propose a general Euler-tour method, using which we show that if a self-similar set satisfies the open set condition and possesses a skeleton, then space-filling curves can be constructed. Especially, all connected selfsimilar sets of finite type fall into this class. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.MSC 2000: 28A80, 54C05.

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