Characterization of infinite LSP words and endomorphisms preserving the LSP property

TL;DR

This paper provides a finite automaton-based characterization of infinite LSP words, showing their structure via $S$-adic representations and identifying endomorphisms that preserve the LSP property.

Contribution

It introduces an $S$-adic characterization of infinite LSP words and characterizes endomorphisms that preserve the LSP property, demonstrating the necessity of automata in their analysis.

Findings

Finite automaton can recognize directive words of LSP words.

No finite set of endomorphisms fully characterizes LSP words.

Automata are essential for recognizing LSP words regardless of morphism set.

Abstract

Answering a question of G. Fici, we give an $S$-adic characterization of thefamily of infinite LSP words, that is, the family of infinite words having all their left special factors as prefixes.More precisely we provide a finite set of morphisms $S$ and an automaton ${\cal A}$ such that an infinite word is LSP if and only if it is $S$-adic and one of its directive words is recognizable by ${\cal A}$.Then we characterize the endomorphisms that preserve the property of being LSP for infinite words.This allows us to prove that there exists no set $S'$ of endomorphisms for which the set of infinite LSP words corresponds to the set of $S'$-adic words. This implies that an automaton is required no matter which set of morphisms is used.