Geometric limits of Julia sets for sums of power maps and polynomials

Abstract

For maps of one complex variable, $f$, given as the sum of a degree $n$ power map and a degree $d$ polynomial, 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 $f$ is the closed unit disk or the unit circle. We also provide a general description, for many cases, of the limiting set.