On Sturmian substitutions closed under derivation

Occurrences of a factor $w$ in an infinite uniformly recurrent sequence ${\bf u}$ can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted ${\bf d_{\bf u}}(w)$ and called the derived sequence to $w$ in ${\bf u}$. If $w$ is a prefix of a fixed point ${\bf u}$ of a primitive substitution $\varphi$, then by Durand's result from 1998, the derived sequence ${\bf d_{\bf u}}(w)$ is fixed by a primitive substitution $\psi$ as well. For a non-prefix factor $w$, the derived sequence ${\bf d_{\bf u}}(w)$ is fixed by a substitution only exceptionally. To study this phenomenon we introduce a new notion: A finite set $M$ of substitutions is said to be closed under derivation if the derived sequence ${\bf d_{\bf u}}(w)$ to any factor $w$ of any fixed point ${\bf u}$ of $\varphi \in M$ is fixed by a morphism $\psi \in M$. In our article we characterize the Sturmian substitutions which belong to a set $M$ closed under derivation. The characterization uses either the slope and the intercept of its fixed point or its S-adic representation.