The sequence of return words of the Fibonacci sequence

Let omega be a factor of the Fibonacci sequence F-infinity = x(1)x(2) ..., then it occurs in the sequence infinitely many times. Let omega(p) be the p-th occurrence of omega and r(p)(omega) be the p-th return word over omega. In this paper, we study the structure of the sequence of return words {r(p)(omega)}(p >= 1). We first introduce the singular kernel word sk(omega) for any factor omega of F-infinity and give a decomposition of co with respect to sk(omega). Using the singular kernel and the decomposition, we prove that the sequence of return words over the alphabet {r(1)(omega), r(2)(omega)} 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. (C) 2015 Elsevier B.V. All rights reserved.