Morse and Hedlund's skew Sturmian words revisited

For any infinite word r over a, b we associate two infinite words min(r), max(r) such that any prefix of min(r) (max(r), respectively) is the lexicographically smallest (great- est, respectively) among the factors of r of the same length. We prove that (min(r); max(r)) = (as; bs) for some infinite word s if and only if r is a proper Sturmian word or an ultimately peri- odic word of a particular form. This result is based on a lemma concerning sequences of infinite words.