Sturmian external angles of primitive components in the Mandelbrot set

TL;DR

This paper introduces a geometric algorithm called the broken line construction for computing periodic Sturmian angles and their landing parameters in the Mandelbrot set, establishing a correspondence with kneading sequences.

Contribution

The paper presents a new combinatorial method to compute and analyze periodic Sturmian angles and their landing points in the Mandelbrot set, including conjugation and binary sequence correspondence.

Findings

The broken line construction effectively computes periodic Sturmian angles.

A one-to-one correspondence between binary expansions and kneading sequences is established.

The method simplifies understanding the structure of primitive components in the Mandelbrot set.

Abstract

In this work we introduce the broken line construction, which is a geometric and combinatorial algorithm that computes periodic Sturmian angles of a given period, yielding the locations of their landing parameters in the Mandelbrot set. An easy to implement method to compute the conjugated angle of a periodic Sturmian angle is also provided. Furthermore, if $\theta$ is a periodic Sturmian angle computed by the broken line construction, then we show the existence of a one-to-one correspondence between its binary expansion and its associated kneading sequence.