Surgery sequences and self-similarity of the Mandelbrot set

TL;DR

This paper introduces a new concept called surgery sequences in rational maps, providing a novel proof of the self-similarity of the Mandelbrot set and Julia sets at Misiurewicz points.

Contribution

It develops an analog of hyperbolic Dehn surgery for rational maps and applies it to prove self-similarity properties of the Mandelbrot set.

Findings

New concept of surgery sequences for rational maps

Elementary proof of self-similarity at Misiurewicz points

Connection between surgery sequences and fractal structures

Abstract

We introduce an analog in the context of rational maps of the idea of hyperbolic Dehn surgery from the theory of Kleinian groups. A surgery sequence is a sequence of postcritically finite maps limiting (in a precise manner) to a postcritically finite map with at least one strictly preperiodic critical orbit. As an application of this idea we give a new and elementary proof of Tan Lei's theorem on the asymptotic self-similarity of Julia Sets and the Mandelbrot Set at Misiurewicz points.