On sets of indefinitely desubstitutable words

TL;DR

This paper investigates the structure of infinite words generated by indefinite desubstitution processes, providing new characterizations of well-known word sets and exploring stability under finite substitution sets.

Contribution

It introduces a generalized notion of stable sets of infinite words, characterizes several classical word sets within this framework, and analyzes endomorphisms preserving episturmian words.

Findings

Characterization of stable sets for various infinite word classes

Identification of stable sets among known word sets like balanced and episturmian words

Description of endomorphisms that preserve episturmian words

Abstract

The stable set associated to a given set S of nonerasing endomorphisms or substitutions is the set of all right infinite words that can be indefinitely desubstituted over S. This notion generalizes the notion of sets of fixed points of morphisms. It is linked to S-adicity and to property preserving morphisms. Two main questions are considered. Which known sets of infinite words are stable sets? Which ones are stable sets of a finite set of substitutions? While bringing answers to the previous questions, some new characterizations of several well-known sets of words such as the set of binary balanced words or the set of episturmian words are presented. A characterization of the set of nonerasing endomorphisms that preserve episturmian words is also provided.