Generalizations of Douady's magic formula

TL;DR

This paper extends Douady's combinatorial formula from the main cardioid to other hyperbolic components of the Mandelbrot set, providing an explicit map relating external rays landing on these components to those on the real axis.

Contribution

It introduces a new explicit piecewise linear map that generalizes Douady's formula to all hyperbolic components of the Mandelbrot set.

Findings

Constructed an explicit piecewise linear map for hyperbolic components

Generalized Douady's formula beyond the main cardioid

Established a correspondence between external rays landing on H and the real axis

Abstract

We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components $H$ of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on $H$ to the set of angles of external rays landing on the real axis.