Balances for fixed points of primitive substitutions

An infinite word defined over a finite alphabet A is balanced if for any pair (ω,ω') of factors of the same length and for any letter a in the alphabet ||ω|a - |ω'|a| ≤ 1, where |ω|a denotes the number of occurrences of the letter a in the word ω. In this paper, we generalize this notion and introduce a measure of balance for an infinite sequence. In the case of fixed points of primitive substitutions, we show that the asymptotic behaviour of this measure is in part ruled by the spectrum of the incidence matrix associated with the substitution. Connections with frequencies of letters and other balance properties are also discussed.

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