  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 functions.
