Border correlation of binary words

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, in other words, β ( w ) = c 0 c 1 ... c n - 1 , where c i = a or b according to whether the ith cyclic shift σ i ( w ) of w is unbordered or bordered. Except for some special cases, no binary word w has two consecutive unbordered conjugates ( σ i ( w ) and σ i + 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 ) , β 2 ( w ) ,... terminates either in b n or in the cycle of conjugates of the word ab k ab k + 1 for n = 2 k + 3 .