On the orbits of automaton semigroups and groups

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