On the Conjugation of Standard Morphisms

Let A={a, b} be an alphabet. An infinite word on A is Sturmian if it contains exactly n+1 distinct factors of length n for every integer n. A morphism f on A is Sturmian if f(x) is Sturmian whenever x is. A morphism on A is Standard if it is an element of the monoid generated by the two elementary morphisms E, which exchanges a and b, and φ, the Fibonacci morphism defined by φ(a)=ab and φ(b)=a. The set of Standard morphisms is a proper subset of the set of Sturmian morphisms. In the present paper, we give a characterization of Sturmian morphisms as conjugates of Standard ones. Sturmian words generated by Standard morphisms are characteristic words. The previous result allows to prove that a morphism f generates an infinite word having the same set of factors as a characteristic word generated by a Standard morphism g if and only if f is a conjugate of g.