Conjugates of characteristic Sturmian words generated by morphisms

Glen, A. (2004) Conjugates of characteristic Sturmian words generated by morphisms. European Journal of Combinatorics, 25 (7). pp. 1025-1037.

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Abstract

This article is concerned with characteristic Sturmian words of slope α and 1-α (denoted by cα and c1-α resp.), 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,r̄] (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.