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

TL;DR

This paper introduces a new method for constructing space-filling curves on self-similar sets using skeletons and edge-to-trail substitution rules, providing an algorithmic approach under certain conditions.

Contribution

It defines skeletons for self-similar sets and develops a general Euler-tour method to construct space-filling curves, unifying classical constructions and extending applicability.

Findings

The Euler-tour method guarantees space-filling curves for sets with skeletons satisfying the open set condition.

Connected self-similar sets of finite type can be constructed using the proposed method.

The approach unifies classical space-filling curve constructions into two main classes.

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.