The solar Julia sets of basic quadratic Cremer polynomials

TL;DR

This paper constructs quadratic Cremer Julia sets with positive area, revealing detailed topological structures and properties of external rays and impressions, advancing understanding of complex dynamics near Cremer points.

Contribution

It demonstrates the existence of positive-area quadratic Cremer Julia sets with specific local connectivity and impression properties, a significant development in complex dynamics.

Findings

Existence of positive-area quadratic Cremer Julia sets.

Full Lebesgue measure set of angles with degenerate impressions.

Julia sets are connected im kleinen at certain landing points.

Abstract

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.