Selfdual Substitutions in Dimension One

TL;DR

This paper investigates various notions of duality in one-dimensional two-letter substitutions, establishing their equivalence and providing arithmetic criteria for selfduality, thereby clarifying the relationships and offering a comprehensive overview.

Contribution

It demonstrates the equivalence of different dual notions for 1D two-letter substitutions and derives necessary and sufficient arithmetic conditions for selfduality.

Findings

Most dual notions are equivalent in 1D two-letter substitutions

Arithmetic conditions characterize selfduality

The paper serves as a survey on substitution dualities

Abstract

There are several notions of the 'dual' of a word/tile substitution. We show that the most common ones are equivalent for substitutions in dimension one, where we restrict ourselves to the case of two letters/tiles. Furthermore, we obtain necessary and sufficient arithmetic conditions for substitutions being selfdual in this case. Since many connections between the different notions of word/tile substitution are discussed, this paper may also serve as a survey paper on this topic.