Hyperbolic polynomial diffeomorphisms of C2C2. III: Iterated monodromy groups

This paper is a sequel to Part I [13] and Part II [14] and [15]. In the current article we relate several combinatorial descriptions of the Julia sets for hyperbolic polynomial diffeomorphisms of C2C2: quotients of solenoids [3], automata [22] and Hubbard trees [14] and [15]. The notion of iterated monodromy groups are defined for such diffeomorphisms and are used to construct automata from Hubbard trees.