Cyclic complexity of words

International audienceWe introduce and study a complexity function on words c x (n), called cyclic complexity, which counts the number of conjugacy classes of factors of length n of an infinite word x. We extend the well-known Morse–Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if x is a Sturmian word and y is a word having the same cyclic complexity of x, then up to renaming letters, x and y have the same set of factors. In particular, y is also Sturmian of slope equal to that of x. Since c x (n) = 1 for some n ≥ 1 implies x is periodic, it is natural to consider the quantity lim inf n→∞ c x (n). We show that if x is a Sturmian word, then lim inf n→∞ c x (n) = 2. We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the ✩ Some of the results in this paper were presented at 37 limit infimum. In contrast we show that, for the Thue–Morse word t, lim inf n→∞ c t (n) = +∞