Given an $\omega$-automaton and a set of word homomorphisms, we look at which accepted words have such a substitutive structure, and in particular if there is at least one. We introduce a method using desubstitution of $\omega$-automata to describe the structure of preimages of accepted words under arbitrary sequences of homomorphisms: this takes the form of a meta-$\omega$-automaton. We decide the existence of an accepted purely substitutive word, as well as the existence of an accepted fixed point. In the case of multiple substitutions (non-erasing homomorphisms), we decide the existence of an accepted infinitely desubstitutable word, with possibly some constraints on the sequence of substitutions (e.g. Sturmian words or Arnoux-Rauzy words). As an application, we decide when a set of finite words codes e.g. a Sturmian word. As another application, we also show that if an $\omega$-automaton accepts a Sturmian word, it accepts the image of the full shift under some Sturmian morphism.

翻译：给定一个 ω-自动机和一组词同态，我们研究哪些被接受词具有此类替换结构，特别关注是否存在至少一个这样的词。我们引入一种基于 ω-自动机去替换的方法，用以描述被接受词在任意同态序列下的原像结构：该结构表现为一个元 ω-自动机。我们判定是否存在被接受的纯替换词，以及是否存在被接受的不动点。在多重替换（非擦除同态）的情形下，我们判定是否存在被接受的无限可替换词，同时可能对替换序列施加某些约束（例如 Sturmian 词或 Arnoux-Rauzy 词）。作为应用，我们判定一组有限词何时编码例如一个 Sturmian 词。另一应用是，我们还证明：若一个 ω-自动机接受某个 Sturmian 词，则它必然接受该词在某个 Sturmian 态射下完全移位像。