The Julia sets of basic uniCremer polynomials of arbitrary degree

TL;DR

This paper classifies Julia sets of certain degree d polynomials with Cremer points into two types, describing their topological properties and how they arise via polynomial-like maps.

Contribution

It introduces a classification of Julia sets for uniCremer polynomials into red dwarf and solar types, detailing their topological structures and formation mechanisms.

Findings

Red dwarf Julia sets are nowhere connected im kleinen with a continuum intersection of impressions.

Solar Julia sets have dense orbit angles with degenerate impressions and are connected im kleinen at landing points.

Such Julia sets appear through polynomial-like maps for generic polynomials with Cremer points.

Abstract

Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The \emph{red dwarf} $J_P$ are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing $p$ and the orbits of all critical images. The \emph{solar} $J_P$ are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and $J_P$ is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of $J_P$ and show that such sets $J_P$ appear through polynomial-like maps for generic polynomials with Cremer points.