Abelian powers and repetitions in Sturmian words

Richomme, Saari and Zamboni (2011) [39] proved that at every position of a Sturmian word starts an abelian power of exponent k   for every k>0k>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 α. By considering all possible abelian periods m, we recover the result of Richomme, Saari and Zamboni.As an analogue of the critical exponent, we introduce the abelian critical exponent A(sα)A(sα) of a Sturmian word sαsα of angle α   as the quantity A(sα)=limsupkm/m=limsupkm′/m, where kmkm (resp. km′) 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(sα)A(sα) equals the Lagrange constant of the number α  . This yields a formula for computing A(sα)A(sα) in terms of the partial quotients of the continued fraction expansion of α  . Using this formula, we prove that A(sα)≥5 and that the equality holds for the Fibonacci word. We further prove that A(sα)A(sα) is finite if and only if α   has bounded partial quotients, that is, if and only if sαsα 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 FjFj, j>1j>1, has length Fj(Fj+1+Fj−1+1)−2Fj(Fj+1+Fj−1+1)−2 if j   is even or Fj(Fj+1+Fj−1)−2Fj(Fj+1+Fj−1)−2 if j   is odd, where FjFj 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: we prove that for j≥3j≥3 the Fibonacci word fjfj, of length FjFj, has minimum abelian period equal to F⌊j/2⌋F⌊j/2⌋ if j=0,1,2mod4 or to F1+⌊j/2⌋F1+⌊j/2⌋ if j=3mod4.