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

TL;DR

This paper introduces a criterion for the existence of skeletons in connected self-similar sets, enabling the construction of space-filling curves through neighbor graph analysis, and provides an algorithm for sets satisfying the finite type condition.

Contribution

It establishes a neighbor graph-based criterion for skeleton existence and offers an algorithm for sets with the finite type condition, advancing the construction of space-filling curves.

Findings

Connected self-similar sets with finite type condition always have skeletons.

A neighbor graph criterion determines skeleton existence.

An algorithm for constructing skeletons is provided.

Abstract

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. It is shown in [Dai, Rao and Zhang, Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule,https://doi.org/10.1088/1361-6544/ab1275] that for a connected self-similar set, space-filling curves can be constructed provided that it possesses a skeleton. 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.