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.Comment: 26 page

Balance and Abelian complexity of the Tribonacci word

G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $R^2$, each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity AC(n) of the Tribonacci word $t$ which is the unique fixed point of $\tau$. We show that $AC(n)\in {3,4,5,6,7}$ for each $n\geq 1$, and that each of these five values is assumed. Our proof relies on the fact that the Tribonacci word is 2-balanced, i.e., for all factors $U$ and $V$ of $t$ of equal length, and for every letter $a \in {0,1,2}$, the number of occurrences of $a$ in $U$ and the number of occurrences of $a$ in $V$ differ by at most 2. While this result is announced in several papers, to the best of our knowledge no proof of this fact has ever been published. We offer two very different proofs of the 2-balance property of $t$. The first uses the word combinatorial properties of the generating morphism, while the second exploits the spectral properties of the incidence matrix of $\tau$.Comment: 20 pages, 1 figure. This is an extended version of 0904.2872v