Master Teapots and Entropy Algorithms for the Mandelbrot Set

TL;DR

This paper extends the concept of the Master teapot to principal veins of the Mandelbrot set, analyzing eigenvalue behavior and core entropy algorithms to reveal geometric and topological properties.

Contribution

It introduces a generalized Master teapot for Mandelbrot principal veins and establishes the equivalence of core entropy algorithms, advancing understanding of complex dynamics.

Findings

Eigenvalues outside the unit circle vary continuously.

Roots inside the unit circle exhibit persistence.

The outside part of the Thurston set is path connected.

Abstract

We construct an analogue of W. Thurston's "Master teapot" for each principal vein in the Mandelbrot set, and generalize geometric properties known for the corresponding object for real maps. In particular, we show that eigenvalues outside the unit circle move continuously, while we show "persistence" for roots inside the unit circle. As an application, this shows that the outside part of the corresponding "Thurston set" is path connected. In order to do this, we define a version of kneading theory for principal veins, and we prove the equivalence of several algorithms that compute the core entropy.