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

## TL;DR

This paper characterizes the geometric limits of Julia sets for maps combining power maps and polynomials as the degree of the power map tends to infinity, identifying conditions for convergence to the unit disk or circle.

## Contribution

It establishes necessary and sufficient conditions for the limits of Julia sets in these maps and describes their structure in many cases.

## Key findings

- Limits are the closed unit disk or circle under certain conditions.
- Provides a comprehensive description of the limiting sets.
- Identifies key parameters influencing the geometric limits.

## 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.