Generalized Thue-Morse words and palindromic richness

TL;DR

This paper investigates the generalized Thue-Morse words, proving their palindromic richness under a dihedral group of symmetries and analyzing their factor complexity, thus extending understanding of their combinatorial structure.

Contribution

The paper introduces the concept of $D_m$-richness for generalized Thue-Morse words and proves their palindromic saturation under all antimorphisms in the dihedral group $D_m$, along with complexity calculations.

Findings

Generalized Thue-Morse words are closed under dihedral group symmetries.

They are saturated by antimorphisms up to the highest possible level.

The factor complexity of these words is explicitly calculated.

Abstract

We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \geq 2$ and $m \geq 1$ as $\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov\'a, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.