Iterated Monodromy Groups of Entire Maps and Dendroid Automata

TL;DR

This paper explores the structure of iterated monodromy groups for transcendental entire functions, introducing dendroid automata to describe their actions and analyzing conditions for amenability.

Contribution

It introduces dendroid automata as a new tool to describe iterated monodromy groups of entire functions, linking automata theory with complex dynamics.

Findings

Iterated monodromy groups can be described by bounded activity automata.

Amenability of these groups is equivalent to the monodromy group's amenability.

Dendroid automata provide a new perspective on the dynamics of entire functions.

Abstract

This paper discusses iterated monodromy groups for transcendental functions. We show that for every post-singularly finite entire transcendental function, the iterated monodromy action can be described by bounded activity automata of a special form, called "dendroid automata". In particular, we conclude that the iterated monodromy group of a post-singularly finite entire function is amenable if and only if the monodromy group is.