Geometrization of sub-hyperbolic semi-rational branched coverings

TL;DR

This paper develops a method to decompose certain complex dynamical systems into simpler parts, proving that each part is equivalent to a unique rational map with specific geometric properties.

Contribution

It introduces a decomposition approach for sub-hyperbolic semi-rational branched coverings using Thurston obstructions, establishing their equivalence to unique rational maps.

Findings

Decomposition of dynamical systems into sub-systems with specific properties

Proof that sub-hyperbolic semi-rational systems have no Thurston obstruction

Establishment of unique geometric finite rational maps for each sub-system

Abstract

Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical systems is either a post-critically finite type branched covering or a sub-hyperbolic semi-rational type branched covering. If a sub-dynamical system is a post-critically finite type branched covering with a hyperbolic orbifold, then it has no Thurston obstruction and is combinatorially equivalent to a unique post-critically finite rational map (up to conjugation by an automorphism of the Riemann sphere) and, more importantly, if a sub-dynamical system is a sub-hyperbolic semi-rational type branched covering with hyperbolic orbifold, we prove in this paper that it has no Thurston obstruction and is CLH-equivalent to a unique geometrically finite rational map (up to conjugation by an automorphism of the Riemann sphere).