Rational Parameter Rays of The Multibrot Sets

TL;DR

This paper establishes a detailed structure theorem for multibrot sets, analyzing the landing behavior of rational parameter rays and bifurcations, using combinatorial and local analytic methods.

Contribution

It provides a comprehensive understanding of the landing patterns of rational rays in multibrot sets, extending previous combinatorial approaches with local analytic techniques.

Findings

Rational parameter rays land at specific points in multibrot sets.

The structure theorem describes bifurcation phenomena in multibrot sets.

Parabolic and Misiurewicz parameters are shown to be landing points of rational rays.

Abstract

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misiurewicz parameters are landing points of rational parameter rays.