Cyclic Complexity of Words

We introduce and study a new complexity function on words, that we call cyclic complexity, which counts the number of conjugacy classes of factors of each given length. We extend the famous MorseHedlund 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 having different slopes. More precisely, we prove that if x is a Sturmian word and y is a word having the same cyclic complexity of x, then y is Sturmian and, up to renaming letters, has the same language of factors of x.