Geometric Limits of Julia Sets with Parameters on the 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.