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Thuston's Teapots and Graph Directed Systems

Chenxi Wu

TL;DR

This work links Thurston's Master Teapot to a Mandelbrot-type set for a parametrized family of graph directed IFS by using Milnor–Thurston kneading theory to certify non-members and to formulate a combinatorial criterion. It constructs GIFS from the Markov decomposition of a postcritically finite tent map $f_\lambda$, with reversed transition graphs and edge maps $x\mapsto zx+2-z$ or $x\mapsto z-zx$, tying slices of the teapot to the GIFS Mandelbrot sets with $C=1$. The authors prove a bi-implication between symbolically defined $λ$-suitable sequences and infinite-path representations, establishing the main theorem that, for $\lambda\in(\sqrt{2},2)$ with PCF tent maps, the slice $T_λ$ intersected with the unit disk equals the corresponding GIFS Mandelbrot set; they also derive asymptotic self-similarity results via Solomyak’s framework. Together, the results bridge unimodal dynamics, kneading theory, and fractal geometry, revealing a deep combinatorial structure underlying the teapot–Mandelbrot correspondence.

Abstract

Thurston's Master Teapot is a geometric object that encodes the entropies of critically periodic unimodal maps. We establish the connection between this object and the "Mandelbrot set" of graph directed iterated function systems previously studied by Solomyak.

Thuston's Teapots and Graph Directed Systems

TL;DR

This work links Thurston's Master Teapot to a Mandelbrot-type set for a parametrized family of graph directed IFS by using Milnor–Thurston kneading theory to certify non-members and to formulate a combinatorial criterion. It constructs GIFS from the Markov decomposition of a postcritically finite tent map , with reversed transition graphs and edge maps or , tying slices of the teapot to the GIFS Mandelbrot sets with . The authors prove a bi-implication between symbolically defined -suitable sequences and infinite-path representations, establishing the main theorem that, for with PCF tent maps, the slice intersected with the unit disk equals the corresponding GIFS Mandelbrot set; they also derive asymptotic self-similarity results via Solomyak’s framework. Together, the results bridge unimodal dynamics, kneading theory, and fractal geometry, revealing a deep combinatorial structure underlying the teapot–Mandelbrot correspondence.

Abstract

Thurston's Master Teapot is a geometric object that encodes the entropies of critically periodic unimodal maps. We establish the connection between this object and the "Mandelbrot set" of graph directed iterated function systems previously studied by Solomyak.
Paper Structure (6 sections, 3 theorems, 9 equations, 1 figure)

This paper contains 6 sections, 3 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graph Directed Iterated Function Systems
  3. The Criteria in [3]
  4. Proof of the Main Theorem
  5. Show that $w$ are all $\lambda$-suitable
  6. Show that lambda-suitable sequences all appear as some w

Key Result

Theorem 2.4

Suppose there is an infinite path $\gamma$ on $G$ of the form $\gamma_0\gamma_1^\infty$ but is not periodic, and $z\in\mathbb{C}$, $|c|<1$ is a simple root of the equation $f_\gamma(z)=C$, such that there are no other path $\gamma'$ on $G$ starting at $v_0$ satisfying $f_{\gamma'}(z)=C$. Then the Ma

Figures (1)

  • Figure 1: The horizontal slice at $\lambda=\sqrt{3}$ intersecting with the unit disc.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Example 2.6
  • Remark 2.7
  • Definition 3.1
  • Definition 3.2
  • ...and 5 more