On prefixal factorizations of words

TL;DR

This paper studies infinite words with prefixal factorizations, introducing a derived word operation, and explores the class of words stable under repeated derivation, connecting it to a coloring problem and characterizing Sturmian words within this class.

Contribution

It defines the class ${ m P}_ ext{infty}$ of words stable under iterative prefixal derivations and links this class to a coloring problem, providing characterizations for Sturmian words.

Findings

${ m P}_ ext{infty}$ contains all words with stable prefixal factorizations.

Potential counterexamples to the coloring conjecture are among non-periodic ${ m P}_ ext{infty}$ words.

A Sturmian word belongs to ${ m P}_ ext{infty}$ iff it is nonsingular.

Abstract

We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization $x= U_0 U_1U_2 \cdots $ where each $U_i$ is a non-empty prefix of $x.$ With each $x\in {\cal P}_1$ one naturally associates a "derived" infinite word $\delta(x)$ which may or may not admit a prefixal factorization. We are interested in the class ${\cal P}_{\infty}$ of all words $x$ of ${\cal P}_1$ such that $\delta^n(x) \in {\cal P}_1$ for all $n\geq 1$. Our primary motivation for studying the class ${\cal P}_{\infty}$ stems from its connection to a coloring problem on infinite words independently posed by T. Brown in \cite{BTC} and by the second author in \cite{LQZ}. More precisely, let ${\bf P}$ be the class of all words $x\in A^\omega$ such that for every finite coloring $\varphi : A^+ \rightarrow C$ there exist $c\in C$ and a factorization $x= V_0V_1V_2\cdots $ with $\varphi(V_i)=c$ for each $i\geq 0.$ In \cite{DPZ} we conjectured that a word $x\in {\bf P}$ if and only if $x$ is purely periodic. In this paper we show that ${\bf P}\subseteq {\cal P}_{\infty},$ so in other words, potential candidates to a counter-example to our conjecture are amongst the non-periodic elements of ${\cal P}_{\infty}.$ We establish several results on the class ${\cal P}_{\infty}$. In particular, we show that a Sturmian word $x$ belongs to ${\cal P}_{\infty}$ if and only if $x$ is nonsingular, i.e., no proper suffix of $x$ is a standard Sturmian word.