# The Shape of Thurston's Master Teapot

## TL;DR

This paper explores the geometric and topological properties of Thurston's Master Teapot and Thurston set for superattracting unimodal maps, revealing their structure, connectivity, and differences from related sets.

## Contribution

It establishes fundamental properties of the Master Teapot, including connectivity and growth behavior, and explains the appearance of gaps in Thurston set approximations.

## Key findings

- The Master Teapot is connected and contains the unit cylinder.
- The intersection with $\\mathbb{D} \\times \\{c\}$ grows monotonically with c.
- The Thurston set differs from that of postcritically finite tent maps.

## Abstract

We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set $\mathbb{D} \times \{c\}$ grows monotonically with $c$. We show that the Thurston set described above is not equal to the Thurston set for postcritically finite tent maps, and we provide an arithmetic explanation for why certain gaps appear in plots of finite approximations of the Thurston set.