Occurrences of a factor w in an infinite uniformly recurrent sequence u can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted ${\mathbf{d}}_{\mathbf{u}}(w)$ and called the derived sequence to w in u. If w is a prefix of a fixed point u of a primitive substitution φ, then by Durand's result from 1998, the derived sequence ${\mathbf{d}}_{\mathbf{u}}(w)$ is fixed by a primitive substitution ψ as well. For a non-prefix factor w, the derived sequence ${\mathbf{d}}_{\mathbf{u}}(w)$ 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 ${\mathbf{d}}_{\mathbf{u}}(w)$ to any factor w of any fixed point u of $\phi \in M$ 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.

无限均匀循环序列u中因子w的出现可以由有限字母上的无限序列编码。此序列通常表示为${\mathbf{d}}_{\mathbf{ü}}（w）$并呼吁衍生序列W¯¯在ü。如果w是原始替换φ的不动点u的前缀，则根据1998年Durand的结果，得出序列${\mathbf{d}}_{\mathbf{ü}}（w）$也由原始替换ψ固定。对于非前缀因子w，派生序列${\mathbf{d}}_{\mathbf{ü}}（w）$仅通过替换来固定。为了研究这种现象，我们引入了一个新的概念：如果导出序列是可替换的，则在导出时将封闭一个有限的替换集合M${\mathbf{d}}_{\mathbf{ü}}（w）$到任何不动点u的任何因数w$\phi \in \mathrm{中号}$ 由态射定 $\psi \in \mathrm{中号}$。在我们的文章中，我们描述了Sturmian替换，它属于在导数下闭合的集合M。表征使用其固定点或其S-adic表示的斜率和截距。