For bounded automatic actions of inverse semigroups the orbit relation is ω-regular, making first-order statements about orbits and actions decidable, including computability of Fatou component encodings for post-critically finite polynomials.
The word and order problems for self-similar and automata groups
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abstract
We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.
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2026 1verdicts
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Automatic actions I. Bounded automata and orbits
For bounded automatic actions of inverse semigroups the orbit relation is ω-regular, making first-order statements about orbits and actions decidable, including computability of Fatou component encodings for post-critically finite polynomials.