Any infinite uniformly recurrent word u can be written as concatenation of a finite number of return words to a chosen prefix w of u. Ordering of the return words to w in this concatenation is coded by derivated word du(w). In 1998, Durand proved that a fixed point u of a primitive morphism has only finitely many derivated words du(w) and each derivated word du(w) is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism ψ lists the morphisms fixing the derivated words of the Sturmian word u = ψ(u). We provide a sharp upper bound on length of the list.

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Circular D0L-systems are those with finite synchronizing delay. We introduce a tool called graph of overhangs which can be used to find the minimal value of synchronizing delay of a given D0L-system. By studying the graphs of overhangs, a general upper bound on the minimal value of a synchronizing delay of a circular D0L-system with a binary uniform morphism is given

Complementary symmetric Rote sequences are binary sequences which have factor complexity C(n) = 2n for all integers n ≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence v which is associated with a standard Sturmian sequence u. We show that any non-empty prefix of v has three return words. We prove that any derivated sequence of v is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that v is primitive substitutive if and only if u is primitive substitutive. Moreover, if the sequence u is a fixed point of a primitive morphism, then all derivated sequences of v are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.

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