Abelian Powers and Repetitions in Sturmian Words

TL;DR

This paper investigates the maximum exponents of abelian powers and repetitions in Sturmian words, introduces the abelian critical exponent, and relates it to the Lagrange constant of the rotation angle, providing formulas and bounds especially for Fibonacci words.

Contribution

It introduces the abelian critical exponent for Sturmian words, linking it to number theory, and provides explicit formulas for abelian powers, especially in Fibonacci words, improving previous results.

Findings

The abelian critical exponent equals the Lagrange constant of the rotation angle.

A formula for the maximum exponent of abelian powers in Sturmian words is provided.

The minimum abelian period in Fibonacci words is a Fibonacci number.

Abstract

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $\alpha$. vAs an analogue of the critical exponent, we introduce the abelian critical exponent $A(s_\alpha)$ of a Sturmian word $s_\alpha$ of angle $\alpha$ as the quantity $A(s_\alpha) = limsup\ k_{m}/m=limsup\ k'_{m}/m$, where $k_{m}$ (resp. $k'_{m}$) denotes the maximum exponent of an abelian power (resp.~of an abelian repetition) of abelian period $m$ (the superior limits coincide for Sturmian words). We show that $A(s_\alpha)$ equals the Lagrange constant of the number $\alpha$. This yields a formula for computing $A(s_\alpha)$ in terms of the partial quotients of the continued fraction expansion of $\alpha$. Using this formula, we prove that $A(s_\alpha) \geq \sqrt{5}$ and that the equality holds for the Fibonacci word. We further prove that $A(s_\alpha)$ is finite if and only if $\alpha$ 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: 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 $F_{j}$ is the $j$th Fibonacci number; ii) The minimum abelian period of any factor is a Fibonacci number. Further, we derive a formula for the minimum abelian periods of the finite Fibonacci words