Space-filling curves of self-similar sets (I): Iterated function systems with order structure

TL;DR

This paper introduces linear graph-directed IFS to systematically construct space-filling curves of self-similar sets, extending known results and setting the foundation for automated algorithms in subsequent works.

Contribution

It defines linear GIFS and links their structure to space-filling curves, providing a new systematic approach for their construction.

Findings

Introduces linear GIFS for space-filling curve construction

Establishes connection between IFS structures and space-filling curves

Lays groundwork for automated algorithms in subsequent papers

Abstract

This paper is the first paper of three papers in a series, which intend to provide a systematic treatment for the space-filling curves of self-similar sets. In the present paper, we introduce a notion of \emph{linear graph-directed IFS} (linear GIFS in short). We show that to construct a space-filling curve of a self-similar set, it is amount to explore its linear GIFS structures. Some other notions, such as chain condition, path-on-lattice IFS, and visualizations of space-filling curves are also concerned. In sequential papers \cite{Dai15} and \cite{RZ14}, we obtain a universal algorithm to construct space-filling curves of self-similar sets of finite type, that is, as soon as the IFS is given, the computer will do everything automatically. Our study extends almost all the known results on space-filling curves.