  We show that an automaton group or semigroup is infinite if and only if it
admits an $\omega$-word (i. e. a right-infinite word) with an infinite orbit,
which solves an open problem communicated to us by Ievgen V. Bondarenko. In
fact, we prove a generalization of this result, which can be applied to show
that finitely generated subgroups and subsemigroups as well as principal left
ideals of automaton semigroups are infinite if and only if there is an $\omega$
-word with an infinite orbit under their action. The proof also shows some
interesting connections between the automaton semigroup and its dual. Finally,
our result is interesting from an algorithmic perspective as it allows for a
reformulation of the finiteness problem for automaton groups and semigroups.
