Conjugates of characteristic Sturmian words generated by morphisms

This article is concerned with characteristic Sturmian words of slope α and 1−α (denoted by cα and c1−α respectively), where α ∈ (0, 1) is an irrational number such that α = [0; 1 + d1, d2, . . . , dn] with dn ≥ d1 ≥ 1. It is known that both cα and c1−α are fixed points of non-trivial (standard) morphisms σ and σ̂, respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words cα and c1−α are generated by the respective morphisms σ and σ̂. For the particular case when α = [0; 2, r] (r ≥ 1), we give a decomposition of each conjugate of cα (and hence c1−α) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism σ by which it is generated. This extends a recent result of Levé and Séébold on conjugates of the infinite Fibonacci word.