A characterization of Sturmian morphisms

Abstract

A morphism is Sturmiau if the image of every Sturmian word is a Sturmian word. Sturmian morphisms appear in number theory in connection with the so-called substitutions of characteristic sequences. A recent account of results in this direction is given by T. C. Brown in [4]. In this paper, we show that in order to test whether a morphism f is Sturmian, it suffices to check whether the single word f(ba2ba2baba2bab) is balanced. This is in fact a strengthening of a result by Mignosi, Sddbold [14]. The decidability is an immediate consequence. We also get a simpler proof of a theorem by Crisp et al. [5] characterizing those irrational numbers for which the characteristic sequence is a fixed point of a (Sturmian) morphism.* Partially supported by the PRC "Math~matiques et Informatique'.

Decimations and sturmian words

“…Among the features of Sturmian words that have been or can be studied without the help of the algebraic définition, let us mention: the description of the palindrome factors [8], the fact that, if s is proper Sturmian, the Rees quotient monoid A*/(A* \Fac(s)) has the weak permutation property P£ (see [11]), the particular case of the Fibonacci word, and above all, two important subjects, the characterization of the morphisms of A* that leave the set of all Sturmian words globally invariant [2], and the properties of the "finite Sturmian words", that is the factors of (infinité) Sturmian words [7,9], AKNOWLEDGEMENT …”

“…Infinité Sturmian words have been studied under various names for a long time (see [2,4,9] for historical notes). They can be defined either algebraically or by combinatorial properties of their factors (the équivalence is proved in [10]).…”

Sturmian images of non Sturmian words and standard morphisms

“…4 give an example of a wide family of words generated by HD0L systems which are not tag-systems (because a Sturmian morphism is never uniform), some of these words being not generated by D0L-systems (Proposition 4.5).…”

Episturmian words: a survey

“…A morphism ϕ is said to be Sturmian if ϕ(s) is Sturmian for any Sturmian word s. The set of Sturmian morphisms over {a, b} is closed under composition, and consequently it is a submonoid of the endomorphisms of {a, b} * . Moreover, it is well known that the monoid of Sturmian morphisms is generated by the three morphisms: (a → ab, b → a), (a → ba.b → a), (a → b, b → a) and that Sturmian morphisms are precisely the morphisms that map Sturmian words onto Sturmian words (see [16,87]). …”

“…The term 'balanced' is relatively new; it appeared in [16,17] (also see [83,Chapter 2]), and the notion itself dates back to [91,37]. In the pioneering work of Morse and Hedlund [91], balanced infinite words over a 2-letter alphabet were called 'Sturmian trajectories' and belong to three classes: aperiodic Sturmian; periodic Sturmian; and infinite words that are ultimately periodic (but not periodic), called skew words.…”