On the Orbits of Automaton Semigroups and Groups

TL;DR

This paper explores the complexity and structure of orbits in automaton semigroups and groups, revealing undecidability results and properties of infinite and finite orbits for various automaton classes.

Contribution

It establishes undecidability of the finiteness problem for automaton groups and analyzes orbit properties for specific subclasses of automata.

Findings

Finiteness problem for automaton groups is undecidable.

Automata with finite orbits on $oldsymbol{ extomega}$-words produce ultimately periodic words.

Decidability issues for infinite orbits and finite orbit existence in certain automaton classes.

Abstract

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.