The sequence of return words of the Fibonacci sequence

Abstract

Let ω be a factor of the Fibonacci sequence F∞=x1x2⋯, then it occurs in the sequence infinitely many times. Let ωp be the p-th occurrence of ω and rp(ω) be the p-th return word over ω. In this paper, we study the structure of the sequence of return words {rp(ω)}p≥1. We first introduce the singular kernel word sk(ω) for any factor ω of F∞ and give a decomposition of ω with respect to sk(ω). Using the singular kernel and the decomposition, we prove that the sequence of return words over the alphabet {r1(ω),r2(ω)} is still a Fibonacci sequence. We also determine the expressions of return words completely for each factor. Finally we introduce the spectrum for studying some combinatorial properties, such as power, overlap and separate of factors.