  Given an $\omega$-automaton and a set of substitutions, we look at which
accepted words can also be defined through these substitutions, 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.
