Amenability of bounded automata groups on infinite alphabets

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first-level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire transcendental functions in holomorphic dynamics.