Abelian Powers and Repetitions in Sturmian Words

Richomme, Saari and Zamboni proved that at every position of an infinite Sturmian word starts an abelian power of exponent $k$, for every positive integer $k$. Here, we improve on this result, studying the maximal exponent of abelian powers and abelian repetitions occurring in infinite Sturmian words. More precisely, we give a formula for computing the maximal exponent of an abelian power of period $m$ occurring in any Sturmian word $s_{\alpha}$ of rotation angle $\alpha$, and a formula for computing the maximal exponent of an abelian power of period $m$ starting at a given position $n$ in the Sturmian word $s_{\alpha,\rho}$ of rotation angle $\alpha$ and initial point $\rho$. Starting from these results, we introduce the abelian critical exponent $\textit{act}$ as the quantity $\textit{act}=\limsup k_{m}/m=\limsup k'_{m}/m$, where $k_{m}$ (resp. $k'_{m}$) denotes the maximal exponent of an abelian power (resp.~of an abelian repetition) of abelian period $m$. We prove that $\textit{act}\geq \sqrt{5}$ for any Sturmian word, and the equality holds for the Fibonacci word. We further prove that $\textit{act}$ is finite if and only if the development in continued fraction of $\alpha$ has bounded partial quotients, that is, if and only if $s_{\alpha}$ is $\beta$-power free for some real number $\beta$. Concerning the infinite Fibonacci word, we prove that: $\textit{i}$) The longest prefix that is an abelian repetition of period $F_j$, $j>1$, has length $F_j( F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or $F_j( F_{j+1}+F_{j-1} )-2$ if $j$ is odd, where $F_{n}$ is the $n$th Fibonacci number; $\textit{ii}$) The smallest abelian period of any factor of the Fibonacci word is a Fibonacci number. From the previous results, we derive the exact formula for the smallest abelian periods of the Fibonacci finite words.