A Note on a Conjecture of Duval and Sturmian Words

We prove a long standing conjecture of Duval in the special case of Sturmian words. Mathematics Subject Classification. 68R15, 37B10. Let U be a nonempty word on a finite alphabet A. A nonempty word B different from U is called a border of U if B is both a prefix and suffix of U. We say U is bordered if U admits a border, otherwise U is said to be unbordered. For example, U = 011001011 is bordered by the factor 011, while 00010001001 is unbordered. An integer 1 ≤ k ≤ n is a period of a word U = U1 . . . Un if and only if for all 1 ≤ i ≤ n − k we have Ui = Ui+k. It is easy to see that k is a period of U if and only if the prefix B of U of length n − k is a border of U or is empty. Let λ(U) denote the smallest period of U . Then U is unbordered if and only if λ(U) is equal to the length of U, that is λ(U) = |U | = n. We further denote by μ(U) the length of the longest unbordered factor of U . Clearly U is unbordered if and only if μ(U) = |U |. In general, for any word U we have λ(U) ≥ μ(U) (cf. Prop. 2.2 of [3]). An interesting question is to ask for which words U does equality hold. In [3] Duval shows that λ(U) = μ(U) whenever |U | ≥ 2λ(U) − 2 (cf. Cor. 4.2 in [3]). These notions extend directly to infinite words. For an infinite word ω, if μ(ω) is finite, then ω is periodic of period μ(ω). (cf. [2] and [4]). In [3] Duval conjectured that: Conjecture 1 (Duval 1981). Let U be an unbordered word on an alphabet A and let W be a word of length 2|U | beginning in U and with the property that each factor of W of length greater than |U | is bordered. Then W has period |U |, i.e., W = UU.