Abstract

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, i.e., β(w) = b₀b₁...b_n-1, where b_i = a or b according to whether the i-th cyclic shift σⁱ(w) 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 ab^kab^k+1 for n = 2k+3.

Keywords: combinatorics on words, border correlation, binary words

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