Polynomial-like dynamics of analytic maps

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics.

We study dynamical properties of a larger class of maps.

Our main result is that, under some natural conditions, a map of this class has a completely invariant compact set if and only if this set is the filled Julia set of a polynomial-like restriction of the map.

We also generalize this result to include maps with non-connected domains of definition.