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arXiv · 1511.05411

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

Xi-rong Dai·Hui Rao·Shu-qin Zhang
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Abstract

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 self-similar sets of finite type fall into this class. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.

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Xi-rong Dai, Hui Rao, Shu-qin Zhang. 2018-04-23. Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule. https://arxiv.org/abs/1511.05411

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