Carrots for dessert

TL;DR

This paper rigorously proves that dyadic carrot fields and decorations of Mandelbrot set copies shrink to points as their denominators grow large, confirming a conjecture related to the structure of the Mandelbrot set.

Contribution

It formulates and proves a precise asymptotic shrinking statement for dyadic carrot fields and decorations within the Mandelbrot set.

Findings

Dyadic carrot fields shrink to points as denominator increases

Decorations of Mandelbrot set copies also shrink to points with diverging denominators

The proof extends to parabolic Mandelbrot set decorations

Abstract

Carrots for dessert is the title of a section of the paper `On polynomial-like mappings' by Douady and Hubbard. In that section the authors define a notion of dyadic carrot fields of the Mandelbrot set M and more generally for Mandelbrot like families. They remark that such carrots are small when the dyadic denominator is large, but they do not even try to prove a precise such statement. In this paper we formulate and prove a precise statement of asymptotic shrinking of dyadic Carrot-fields around M. The same proof carries readily over to show that the dyadic decorations of copies M' of the Mandelbrot set M inside M and inside the parabolic Mandelbrot set shrink to points when the denominator diverge to infinity.