A Note on ℓ-Rauzy Graphs for the Infinite Fibonacci Word

The ℓ-Rauzy graph of order k for any infinite word is a directed graph in which an arc (v_1,v_2) is formed if the concatenation of the word v_1 and the suffix of v_2 of length k-ℓ is a subword of the infinite word. In this paper, we consider one of the important aperiodic recurrent words, the infinite Fibonacci word for discussion. We prove a few basic properties of the ℓ-Rauzy graph of the infinite Fibonacci word. We also prove that the ℓ-Rauzy graphs for the infinite Fibonacci word are strongly connected.