Homeomorphisms between limbs of the Mandelbrot set

TL;DR

This paper proves that limbs of the Mandelbrot set with equal denominators are homeomorphic, confirming a long-standing conjecture and deepening understanding of the set's structure.

Contribution

It establishes a homeomorphism between limbs of the Mandelbrot set with equal denominators, preserving hyperbolic component periods, thus resolving a conjecture from 1994.

Findings

Limb homeomorphisms with equal denominators are proven to exist.

The homeomorphism preserves periods of hyperbolic components.

The result confirms a longstanding conjecture about the Mandelbrot set.

Abstract

We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994.