Polynomial-like dynamics of analytic maps

TL;DR

This paper extends the theory of polynomial-like maps in holomorphic dynamics by characterizing invariant sets and generalizing to maps with non-connected domains, revealing new structural insights.

Contribution

It introduces a broader class of maps and establishes a criterion for invariant sets being filled Julia sets, including non-connected domain cases.

Findings

Invariant compact sets correspond to filled Julia sets of polynomial-like restrictions.

Generalization to maps with non-connected domains of definition.

Provides a new framework for analyzing complex dynamical systems.

Abstract

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.