Polynomial-like dynamics of analytic maps

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.

Remarks 1.5-1.6 (about related results) and Lemma C.4 added; other minor changes