Border correlation of binary words

The border correlation function β: A* → A*, for A = {a, b}, specifies which conjugates (cyclic shifts) of a given word w of length n are bordered, in other words, β(w) = c0c1...cn-1 where ci=a or b according to whether the ith cyclic shift σi (ω) of w is unbordered or bordered. Except for some special cases, no binary word w has two consecutive unbordered conjugates (σi (w) and σi+1 (w)). We show that this is optimal: in every cyclically overlap-free word every other conjugate is unbordered. We also study the relationship between unbordered conjugates and critical points, as well as, the dynamic system given by iterating the function β. We prove that, for each word w of length n, the sequence w,β(w),β2(w),... terminates either in bn or in the cycle of conjugates of the word abkabk+1 for n = 2k + 3.