Extremal words in morphic subshifts

Given an infinite word x over an alphabet A , a letter b occurring in x , and a total order σ on A , we call the smallest word with respect to σ starting with b 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 ( f ω ( a ) ) for some morphisms f and g , we give two simple conditions on f and g 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.