Geometric Limits of Julia Sets with Parameters on the Circle

TL;DR

This paper investigates the geometric limits of Julia sets for polynomial maps with parameters on the unit circle, revealing non-existence of limits in general but identifying subsequences converging to the unit circle.

Contribution

It demonstrates the non-existence of the geometric limit of filled Julia sets for almost all parameters on the circle and identifies subsequences where the limit is the unit circle.

Findings

Limit of filled Julia sets does not exist for almost all parameters.

A subsequence exists where the limit is the unit circle.

For certain parameters, Julia sets converge to the unit circle.

Abstract

We show that the geometric limit as $n \rightarrow \infty$ of the filled Julia sets $K(P_{n,c})$ for the maps $P_{n,c}(z) = z^n + c$ does not exist for almost every $c$ on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle, and this is used to show that for certain parameters, the geometric limit of the Julia sets $J(P_{n,c})$ is the unit circle.