Lyndon words and Fibonacci numbers

It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound ⌈log2(n)⌉+1⌈log2(n)⌉+1 for the number of distinct Lyndon factors that a Lyndon word of length n   must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is ⌈logϕ(n)⌉+1⌈logϕ(n)⌉+1, where ϕ   denotes the golden ratio (1+5)/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping LxLx that counts the number of Lyndon factors of length at most n in an infinite word x. We show that a recurrent infinite word x is aperiodic if and only if Lx⩾LfLx⩾Lf, where f is the Fibonacci infinite word, with equality if and only if x is in the shift orbit closure of f.