Fibonacci numbers and words

Abstract Let Φ be the golden ratio (√5 + 1)/2, f n the n th Fibonacci finite word and f the Fibonacci infinite word. Let r be a rational number greater than (2 + Φ )/2 and u a nondashempty word. If u r is a factor of f , then there exists n ⩾ 1 such that u is a conjugate of f n and, moreover, each occurrence of u r is contained in a maximal one of ( f n ) s for some s ∈ [2, 2 + Φ ). Several known results on the Fibonacci infinite word follow from this.