Automatic actions I. Bounded automata and orbits

Abstract

We develop the theory of "automatic actions": (semi)groups acting by $\omega$-regular transformations on an $\omega$-regular language, showing that it covers a large class of heretofore-unrelated examples. We focus on the subclass of actions by "bounded" $\omega$-regular transformations, those for which the B\"uchi automata encoding the action do not have two connected non-trivial cycles. We show that, for bounded actions of inverse semigroups, the orbit relation is also $\omega$-regular. We deduce a number of corollaries, in particular decidability, for such actions, of minimality, topological transitivity, aperiodicity, and order of elements. More generally, every first-order statement over the space of the action, involving the action of specific semigroup elements as well as the relation "being in the same orbit", is decidable. We also apply this result to the study of Julia sets of post-critically finite polynomials, and show that the encoding of Fatou components is also computable; thus every first-order statement involving intersection, disjointness etc. of Fatou components or their full orbit under the polynomial, is decidable.