Locally connected models for Julia sets

TL;DR

This paper constructs a finest monotone map from a polynomial Julia set to a locally connected continuum, extending it to a topological polynomial and providing criteria for non-trivial models, thus generalizing key results in complex dynamics.

Contribution

It introduces a universal monotone map for connected Julia sets, extending it to a topological polynomial, and broadens the applicability of existing models in complex dynamics.

Findings

Existence of a finest monotone map onto a locally connected continuum.

Extension of the map to a topological polynomial on the plane.

Criteria for the map not to collapse the Julia set into a point.

Abstract

Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a \emph{finest monotone map $\ph$ onto a locally connected continuum $J_{\sim_P}$}, i.e. a monotone map $\ph:J\to J_{\sim_P}$ such that for any other monotone map $\psi:J\to J'$ there exists a monotone map $h$ with $\psi=h\circ \ph$. Then we extend $\ph$ onto the complex plane $\C$ (keeping the same notation) and show that $\ph$ monotonically semiconjugates $P|_{\C}$ to a \emph{topological polynomial $g:\C\to \C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $\ph$ not to collapse $J$ into a point.