Locating factors of the infinite Fibonacci word

Abstract

Let τ = ( 5 - 1 ) / 2 . Let a , b be two distinct letters. The infinite Fibonacci word is the infinite word G = babbababbabbababbababbabba ⋯ whose nth letter is a (resp., b) if [ ( n + 1 ) τ ] - [ n τ ] = 0 (resp., 1). For a factor w of G, the location of w is the set of all positions in G at which w occurs. Only the locations of the following factors of G are already known: squares, singular words and those factors of G whose lengths are Fibonacci numbers. The purpose of this paper is to determine the locations of all factors of G. Our results contain all the known ones as consequences. Moreover, using our results, we are able to identify any factor of G whenever its starting position and length are given; also we are able to tell whether two suffixes of G have a common prefix of a certain length.