Abelian powers and repetitions in Sturmian words

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79–95, 2011) 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 (an abelian repetition is the analogous of a fractional power in the abelian setting) 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 sα of rotation angle α, 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 sα,ρ of rotation angle α and initial point ρ. Starting from these results, we introduce the abelian critical exponent act as the quantity act = lim sup km/m = lim sup k ′ m/m, where km (resp. k ′ m) denotes the maximal exponent of an abelian power (resp. of an abelian repetition) of abelian period m (in fact, the two superior limits above coincide for every Sturmian word). We prove that act ≥ √ 5 for any Sturmian word, and the equality holds for the Fibonacci word. We further prove that act is finite if and only if the development in continued fraction of α has bounded partial quotients, that is, if and only if sα is β-power free for some real number β. We leave open the question to determine the exact value of act, when it is finite, as a function of the partial quotients of α. Concerning the infinite Fibonacci word, we prove that: i) The longest prefix that is an abelian repetition of period Fj , j > 1, has length Fj(Fj+1 + Fj−1 + 1)− 2 if j is even or Fj(Fj+1 + Fj−1)− 2 if j is odd, where Fn is the nth Fibonacci number; 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. More precisely, we prove that for j ≥ 3, the Fibonacci word fj , of length Fj , has smallest abelian period equal to Fbj/2c if j = 0, 1, 2 mod 4, or to F1+bj/2c if j = 3 mod 4.