A1 Refereed original research article in a scientific journal
Border correlation of binary words
Authors: Harju T, Nowotka D
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
Publication year: 2004
Journal:: Journal of Combinatorial Theory, Series A
Journal name in source: JOURNAL OF COMBINATORIAL THEORY SERIES A
Journal acronym: J COMB THEORY A
Volume: 108
Issue: 2
First page : 331
Last page: 341
Number of pages: 11
ISSN: 0097-3165
DOI: https://doi.org/10.1016/j.jcta.2004.07.009
Abstract
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.
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.
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