On k-abelian Equivalence and Generalized Lagrange Spectra
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We study the set of k-abelian critical exponents of all Sturmian words. It has been proven that in the case k = 1 this set coincides with the Lagrange spectrum. Thus the sets obtained when k > 1 can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when k > 1 the spectrum consists of a dense initial part and a half-line. This is in contrast with the case k=1, where the spectrum contains a half-line, but the initial part is not dense. We describe explicitly the least accumulation points of the generalized spectra and give upper bounds on the left endpoint of the half-lines.
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