2014/02/24 by Cassaigne, Julien, Fici, Gabriele, Sciortino, Marinella +1 · 2 citations
#68R15 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL)
paper · doi:10.48550/arxiv.1402.5843
We introduce and study a complexity function on words cx(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 cx(n)=1 for some n≥ 1 implies x is periodic, it is natural to consider the quantity \liminfn→ ∞ cx(n). We show that if x is a Sturmian word, then \liminfn→ ∞ cx(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 limit infimum. In contrast we show that, for the Thue-Morse word t, \liminfn→ ∞ ct(n)=+∞.