Non-recurrent parameter rays of the Mandelbrot set

TL;DR

This paper establishes a correspondence between non-recurrent parameter rays and non-recurrent parameters in the Mandelbrot set, showing that each non-recurrent angle or parameter is uniquely associated with characteristic angles of the quadratic polynomial.

Contribution

It proves a bi-conditional relationship between non-recurrent parameter rays and non-recurrent parameters in the Mandelbrot set, clarifying the landing behavior of these rays.

Findings

Non-recurrent parameter rays land at non-recurrent parameters.

Each non-recurrent parameter is associated with one or two non-recurrent characteristic angles.

The angles at which these rays land are exactly the characteristic angles of the polynomial.

Abstract

In this paper, we prove that any parameter ray at a non-recurrent angle $\theta$ lands at a non-recurrent parameter $c$ with $\theta$ a characteristic angle of $f_c$; and conversely, every non-recurrent parameter $c$ is the landing point of one or two parameter rays at non-recurrent angles, and these angles are exactly the characteristic angles of $f_c$.