  We investigate the orbits of automaton semigroups and groups to obtain
algorithmic and structural results, both for general automata but also for some
special subclasses. First, we show that a more general version of the
finiteness problem for automaton groups is undecidable. This problem is
equivalent to the finiteness problem for left principal ideals in automaton
semigroups generated by complete and reversible automata. Then, we look at
$\omega$-word (i.e. right infinite words) with a finite orbit. We show that
every automaton yielding an $\omega$-word with a finite orbit already yields an
ultimately periodic one, which is not periodic in general, however. On the
algorithmic side, we observe that it is not possible to decide whether a given
periodic $\omega$-word has an infinite orbit and that we cannot check whether a
given reversible and complete automaton admits an $\omega$-word with a finite
orbit, a reciprocal problem to the finiteness problem for automaton semigroups
in the reversible case. Finally, we look at automaton groups generated by
reversible but not bi-reversible automata and show that many words have
infinite orbits under the action of such automata.
