The Decoration Theorem for Mandelbrot and Multibrot Sets

TL;DR

This paper proves the decoration theorem for Mandelbrot and Multibrot sets, showing that removing small copies results in components with small diameters, enhancing understanding of their fractal structure.

Contribution

It establishes the decoration theorem for Mandelbrot and Multibrot sets, a significant advancement in fractal geometry and complex dynamics.

Findings

Most connected components after removing small Mandelbrot sets have small diameters

The theorem applies to both Mandelbrot and Multibrot sets

Provides a detailed geometric understanding of the fractal structure

Abstract

We prove the decoration theorem for the Mandelbrot set (and Multibrot sets) which says that when a "little Mandelbrot set" is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.