On gapped repeats in a cyclic Fibonacci word

Abstract

In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $({\hat{I}}^1,\hat{I}º)$ of indices such that the word of length $s$ from ${\hat{I}}^1$ is equal to the word of length $s$ from $\hat{I}º$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.