On the conjugation of standard morphisms

Sturmian images of non Sturmian words and standard morphisms

“…Since x is generated by g, this is a direct consequence of [17], Theorem 7 which states that all the Sturmian words generated by morphisms are rigid (see also [14] for a complete proof of this theorem).…”

“…But, since every Sturmian word is recurrent ( [10], Proposition 6.1.2), wc appears as a factor of x in another position, which means thatcwc is also a factor of x. Now, since g = Id A , g ∈ St implies that one of the two words g(c) or g(c) is a proper prefix or a proper suffix of the other one ( [17], Lemma 3). But g(c) = c and g(c) begins withc, thus g(c) is a proper suffix of g(c) which means that g(c) ends with c.…”

“…In Séébold, 1998 [17] it is proved that each Sturmian morphism is a conjugate of a unique standard morphism obtained by replacing, in any writing of the morphism, all the occurrences ofφ by ϕ. This construction is important because many properties of the morphism can be deduced from that of its standard representative.…”

“…They are representatives of the whole Sturmian morphisms thanks to the following conjugacy operation (see [8] and [17]). …”

“…These words and morphisms have been widely studied (see [8] for a general overview) and the Sturmian morphisms are very well known, especially those which generate Sturmian words (see, e.g., [17]). However words generated by morphisms are only a particular case of words generated by D0L systems and, even more generally, by HD0L systems which are special subfamilies of Lindenmayer systems, a large class of mechanisms to produce infinite words which has been the object of hundreds of papers.…”