Lyndon words and Fibonacci numbers

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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 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 logphi(n) + 1, where phi denotes the golden ratio (1 + sqrt5)/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping Lₓ 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 >= Lf, where f is the Fibonacci infinite word, with equality if and only if f is in the shift orbit closure of f.