Orbits of Automaton Semigroups and Groups.

We study the orbits of right infinite or $\omega$-words under the action of semigroups and groups generated by automata. We see that an automaton group or semigroup is infinite if and only if it admits an $\omega$-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. We also discuss the situation in self-similar semigroups and groups and present some applications of the result. Additionally, we investigate the orbits of periodic and ultimately periodic words as well as the existence of $\omega$-words whose orbit is finite.