Geometric limits of Mandelbrot and Julia sets under degree growth

TL;DR

This paper investigates the geometric limits of Mandelbrot and Julia sets for families of polynomial and rational maps as their degree tends to infinity, revealing convergence to the unit disk and circle.

Contribution

It establishes the existence of geometric limits for these sets under degree growth and extends results to generalized families of maps.

Findings

Mandelbrot sets converge to the unit disk as degree increases.

Julia sets tend to the unit circle in the limit.

Results apply to polynomial and rational map families.

Abstract

First, for the family P_{n,c}(z) = z^n + c, we show that the geometric limit of the Mandelbrot sets M_n(P) as n tends to infinity exists and is the closed unit disk, and that the geometric limit of the Julia sets J(P_{n,c}) as n tends to infinity is the unit circle, at least when the modulus of c is not one. Then we establish similar results for some generalizations of this family; namely, the maps F_{t,c} (z) = z^t+c for real t>= 2, and the rational maps R_{n,c,a} (z) = z^n + c + a/z^n.