α-Words and factors of characteristic sequences

✍ Scribed by Wai-Fong Chuan

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 731 KB
• Volume: 177
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(96)00355-x

✦ Synopsis

Let a be an irrational number with 0 < a < 1. Using the continued fraction expansion of a, the class of a-words is introduced. It contains certain sequences of words that are known to relate to the characteristic sequence f(a) of a. When a = (v'~-1)/2, a-words are precisely the Fibonacci words. In this paper, the class of a-words is shown to be a subset of factors of f(a), which is closed under both conjugation and reversion. The canonical palindrome factorization of unbordered a-words play an important role in the determination of factors of f(a). It is proved that every unbordered a-word w that we obtain determines a (Iwl+l)× [wl matrix C of the form 1 such that for every 1 ~< k ~<lwl, the rows of the upper left (k+ 1)× k submatrix are distinct factors of f(e) of length k. As a consequence of a well-known result, this actually gives all the factors of f(a) of length k.