Limits of Julia Sets for Sums of Power Maps and Polynomials

Abstract

Suppose f_{n,c} is a complex-valued mapping of one complex variable given by f_{n,c}(z) = z^n + p(z) + c, where p is a polynomial such that p(0) = 0 and c is a complex parameter such that |c| \u3c 1. We provide necessary and sufficient conditions that the geometric limit, as n approaches infinity, of the set of points that remain bounded under iteration by f_{n,c} is the disk of radius 1 centered at the origin