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

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.