Extremal words in morphic subshifts

Given an infinite word x over an alphabet AA, a letter bb occurring in x, and a total order σσ on AA, we call the smallest word with respect to σσ starting with bb in the shift orbit closure of x an extremal word   of x. In this paper we consider the extremal words of morphic words. If x=g(fω(a)) for some morphisms ff and gg, we give two simple conditions on ff and gg that guarantee that all extremal words are morphic. This happens, in particular, when x is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the period-doubling word and the Chacon word and a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin–Shapiro word.