Palindromic closures using multiple antimorphisms

TL;DR

This paper characterizes when generalized Thue--Morse words are also generalized pseudostandard words, revealing they are either periodic or when the base is less than or equal to the modulus, extending prior results.

Contribution

It extends the characterization of generalized pseudostandard words to include generalized Thue--Morse words, identifying conditions for their equivalence.

Findings

Generalized Thue--Morse words are pseudostandard if and only if they are periodic or b ≤ m.

This extends previous results from classical to generalized Thue--Morse words.

Provides a complete characterization linking periodicity and the parameters b and m.

Abstract

Generalized pseudostandard word $\bf u$, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet ${\cal A}$ and by a directive sequence of involutory antimorphisms acting on ${\cal A}^*$. Prefixes of $\bf u$ with increasing length are constructed using pseudopalindromic closure operator. We show that generalized Thue--Morse words ${\bf t}_{b,m}$, with $b, m \in \N$ and $b, m \geq 2$, are generalized pseudostandard words if and only if ${\bf t}_{b,m}$ is a periodic word or $b \leq m$. This extends the result of de Luca and De Luca obtained for the classical Thue--Morse words.