Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words

TL;DR

This paper investigates a specific class of infinite ternary words generated by a family of substitutions, demonstrating they are 3-balanced and have Abelian complexity bounded by 7, with these bounds proven to be optimal.

Contribution

It introduces a new class of infinite ternary words and establishes their balance and Abelian complexity bounds, which are shown to be tight.

Findings

The fixed points of the substitution are 3-balanced.

Their Abelian complexity is bounded above by 7.

The bounds are proven to be optimal.

Abstract

A word $u$ defined over an alphabet $\mathcal{A}$ is $c$-balanced ($c\in\mathbb{N}$) if for all pairs of factors $v$, $w$ of $u$ of the same length and for all letters $a\in\mathcal{A}$, the difference between the number of letters $a$ in $v$ and $w$ is less or equal to $c$. In this paper we consider a ternary alphabet $\mathcal{A}=\{L,S,M\}$ and a class of substitutions $\phi_p$ defined by $\phi_p(L)=L^pS$, $\phi_p(S)=M$, $\phi_p(M)=L^{p-1}S$ where $p>1$. We prove that the fixed point of $\phi_p$, formally written as $\phi_p^\infty(L)$, is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of $p$. We also show that both these bounds are optimal, i.e. they cannot be improved.