### Abelian Powers and Repetitions in Sturmian Words [article]

Gabriele Fici, Alessio Langiu, Thierry Lecroq, Arnaud Lefebvre, Filippo Mignosi, Jarkko Peltomäki, Élise Prieur-Gaston
2016 arXiv   pre-print
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
more » ... tation angle α. vAs an analogue of the critical exponent, we introduce the abelian critical exponent A(s_α) of a Sturmian word s_α of angle α as the quantity A(s_α) = 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_α) equals the Lagrange constant of the number α. This yields a formula for computing A(s_α) in terms of the partial quotients of the continued fraction expansion of α. Using this formula, we prove that A(s_α) ≥√(5) and that the equality holds for the Fibonacci word. We further prove that A(s_α) is finite if and only if α has bounded partial quotients, that is, if and only if s_α is β-power-free for some real number β. 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 jth 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