Conjugates of characteristic Sturmian words generated by morphisms

Abstract

This article is concerned with characteristic Sturmian words of slope α and 1− α (denoted by c α and c 1− α resp.), where α∈(0,1) is an irrational number such that α=[0;1+d 1, d 2,…,d n ] with d n ≥ d 1≥1. It is known that both c α and c 1− α are fixed points of non-trivial (standard) morphisms σ and σ ̂ , respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words c α and c 1− α are generated by the respective morphisms σ and σ ̂ . For the particular case when α=[0;2, r ̄ ] (r≥1) , we give a decomposition of each conjugate of c α (and hence c 1− 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.