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

This paper is the first part of a series which provides a systematic treatment of the space-filling curves of self-similar sets. In the present paper, we introduce a notion of linear graph-directed IFS (linear GIFS in short). We show that to construct a space-filling curve of a self-similar set, it amounts to exploring its linear GIFS structures. Compared to the previous methods, such as the L-system or recurrent set method, the linear GIFS approach is simpler, more rigorous and leads to further studies on this topic. We also propose a new algorithm for the beautiful visualization of space-filling curves.
In a series of papers Dai et al (2015 arXiv:1511.05411 [math.GN]), Rao and Zhang (2015) and Rao and Zhang (2015), we investigate for a given self-similar set how to get 'substitution rules' for constructing space-filling curves, which was obscure in the literature. We solve the problem for self-similar sets of finite type, which covers most of the known results on constructions of space-filling curves.