Renormalizations and wandering Jordan curves of rational maps

TL;DR

This paper introduces a dynamical decomposition for certain rational maps, revealing wandering Jordan curves and renormalizations, and presents a surgery method to construct such maps with specific properties.

Contribution

It provides a new decomposition framework for post-critically finite rational maps and a surgery technique to generate maps with wandering Jordan curves and prescribed renormalizations.

Findings

Decomposition of the Riemann sphere into invariant subsets with Jordan curves.

Existence of wandering Jordan curves in the Julia set.

A surgery procedure to produce rational maps with desired dynamical features.

Abstract

We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of uncountably many Jordan curve components. Most of them are wandering. The other consists of components that are pullbacks of finitely many renormalizations, together with possibly uncountably many points. The quotient action on the decomposed pieces is encoded by a dendrite dynamical system. We also introduce a surgery procedure to produce post-critically finite rational maps with wandering Jordan curves and prescribed renormalizations.