Border correlation of binary words

: Harju T, Nowotka D

Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE

: 2004

: Journal of Combinatorial Theory, Series A

: JOURNAL OF COMBINATORIAL THEORY SERIES A

: J COMB THEORY A

: 108

: 2

: 331

: 341

: 11

: 0097-3165

DOI: https://doi.org/10.1016/j.jcta.2004.07.009

The border correlation function beta: A* --> A*, for A = {a, b}, specifies which conjugates (cyclic shifts) of a given word w of length n are bordered, in other words, beta(w) = c(0)c(1)... c(n- 1), where c(i) = a or b according to whether the ith cyclic shift sigma(i)(w) of w is unbordered orbordered. Except for some special cases, no binary word w has two consecutive unbordered conjugates (sigma(i)(w) and sigma(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 beta. We prove that, for each word w of length n, the sequence w, beta(w), beta(2)(w),... terminates either in b(n) or in the cycle of conjugates of the word ab(k)ab(k+l) for n = 2k + 3. (C) 2004 Elsevier Inc. All rights reserved.

Mathematics