Abstract:
Given an infinite word x over an alphabet A, a letter b occurring in
x, and a total order \sigma on A, we call the smallest word with respect to \sigma
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^\omega(a))
for some morphisms f and g, we give two simple conditions on f and
g that guarantees 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 give a new proof of
the form of the lexicographically least word in the shift orbit closure of
the Rudin-Shapiro word.
