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Thurston theory

In the 1980’s, Bill Thurston developed a unified theory of geometry of 3-manifolds, automorphisms of closed surfaces, and iterated rational maps. For geometry, he had the vision that every 3-manifold should be geometric, i.e. that it can be decomposed in a natural way so that each component carries a standard geometric structure, and he described the necessary eight standard geometries. His vision was completed more recently in Grigori Perelman’s celebrated work that included the Poincaré conjecture as a special case. Thurston himself realized that by far the most ubiquitous geometry was hyperbolic geometry, and proved several hyperbolization theorems for 3-manifolds.

For closed surfaces, he gave a classification of automorphisms: if $f: S\to S$ is a homeomorphism from a surface $S$ to itself, then (up to homotopy) it is either periodic, reducible, or it admits an invariant geometric structure called a pseudo-Anosov structure.

Finally, for iterated rational maps, his fundamental characterization theorem says that if $f:\mathbb{S}^2\to \mathbb{S}^2$ is a topological branched cover in which the branch points have finite orbits, then this map is realized as a holomorphic map (in a precise sense called Thurston equivalence) if and only if $f$ does not admit a particular kind of obstruction.

Thurston treated all three cases in a unified way: he starts with a topological object and associates to it an iteration procedure in a finite dimensional Teichmüller space. If this iterated map $\sigma$ has a fixed point in Teichmüller space, then the topological problem carries the desired geometry (the topological manifold admits a hyperbolic structure, the surface automorphism admits a pseudo-Anosov structure, and the branched cover admits an invariant complex structure). If $\sigma$ does not have a fixed point, then there is a combinatorial-topological obstruction that can be classified (or excluded by hypothesis in the theorems). All three theorems play fundamental roles in their respective areas of mathematics, and they present deep connections between apparently different areas of mathematics.

Within holomorphic dynamics, Thurston’s theorem is one of the most fundamental theorems we have, and it is underlying essentially all classification theorems of rational maps, including polynomials and Newton maps. With Hubbard and Shishikura, we developed a version of Thurston’s theorem for the simplest kind of transcendental maps, exponential maps, and current work in progress extends this to larger classes of transcendental maps with finite singular orbits. For certain families of maps, we are even working on extending Thurston theory to the “postsingularly infinite” case, which involves iteration in infinite dimensional Teichmüller spaces.

Some of our main results:

• A simplified version of the Thurston map in Teichmüller space, known as “spiders”, that applies to entire functions (polynomials and transcendental maps):
John Hubbard and Dierk Schleicher: The spider algorithm. Complex dynamical systemsAmer. Math. Soc. (1994), 155-180. (article)
• John Hubbard, Dierk Schleicher, and Mitsuhiro Shishikura: Exponential Thurston maps and limits of quadratic differentials. Journal of the American Mathematical Society, 22 (2009), 77-117. (article)
• A study of the action of the Thurston mapping to the boundary of Teichmüller space":
Nikita Selinger: Thurston’s pullback map on the augmented Teichmüller space and applications. Inventiones Mathematicae, 189 (2012), 111-142. (arxiv)

Experts within our team: Wolf Jung, Bayani Hazemach, Mikhail Hlushchanka, Dierk Schleicher.