Space-filling curves of self-similar sets (II): edge-to-trail substitution rule*The work is supported by CNFS Nos 11431007, 11771457, and Project I3346 and the doctoral program W1230 granted by the Austrian Science Fund (FWF). (12th April 2019)

• Record Type: Journal Article
• Authors: Dai, Xin-Rong; Rao, Hui; Zhang, Shu-Qin
• Is Part Of: Nonlinearity. Volume 32:Number 5(2019:May)
• Page Start: 1772
• Page End: 1809
• Publication Date: 2019-04-12
• Subjects: space-filling curves -- self-similar sets -- edge-to-trail substitution; 28A80 -- 54C05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab1275
• Languages: English
• ISSNs: 0951-7715

Abstract

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 this method 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 provides an algorithm to construct space-filling curves of self-similar sets.