Math 885501 Fall 2018
Topics in Geometry & Topology
Topic 1: Entropy and pseudo-Anosov maps
Topic 2: Thurston-Milnor kneading theory and related topics
The core text for our exploration of this topic.
Classifies numbers which arise as the topological entropy associated to
postcritically finite maps of the unit interval. The manuscript is
full of intriguing ideas and questions, but the exposition is not very
polished and is hard to read. I'll talk about the first few sections;
we may return to subsequent sections later in the course.
Thurston's notion of "core entropy"
extends ideas from Milnor-Thurston kneading theory to the realm of
holomorphic dynamics and, in particular, Julia sets. Tiozzo
proves that core entropy of quadratic polynomials is a continuous
function of external angle in parameter space.
A post by Thurston in response to a question on math overflow. The Schwarzian derivative comes up in On iterated maps of the interval; I'd like to understand this post and how it relates to maps of the unit interval.
Topic 3: Thurston's topological characterization of rational maps
Our main text for this section.
We broaden our focus from postcritically finite maps of the unit
interval to postcriticallyfinite maps of the Riemann sphere.
Topic 4: Matings
- Thurston equivalence of topological polynomials, by Laurent Bartholdi and Volodymyr Nekrashevych
Topic 5: Shapes of polyhedra and triangulations of the sphere