Thurston theory in transcendental dynamics

在20世纪80年代，Bill Thurston发展了一个关于3 $-流形、曲面自同构和迭代有理映射的统一理论;所得到的定理都依赖于有限维Teichmüller空间中的迭代过程。特别是对于有理映射，他的定理在用符号动力学描述迭代映射方面具有根本的重要性：在结构重要的后临界有限映射的情况下，不变复结构的刚性使得有可能提取足够多的组合不变量来区分不同的映射，并且全纯映射的分类基于Thurston定理。在每一个特定的情况下，找到和分类适当的组合不变量本身就是一个定理。这样的不变量已经被发现并分类为迭代多项式（哈伯德树和批判肖像），但不是一般的理性映射（除了作为多项式的牛顿映射出现的理性映射）。大约30年来，人们一直在努力将瑟斯顿定理从有理映射推广到超越映射。唯一可用的“超越”推广瑟斯顿的定理，由哈伯德，志仓，和提议者，只涵盖了简单的家庭指数映射。然而，它为扩展到更一般的超越映射奠定了基础。描述多项式动力学的所有基本概念（动态射线，哈伯德树，蜘蛛，关键的肖像，等等），不容易获得的先验函数，但最近取得了重大进展，在相当大的一般性，例如，在证明动态射线（又名“头发”）的存在，为许多家庭的地图。这个项目的目的是扩展瑟斯顿的基本定理，并由此成功分类的后临界有限多项式，超越世界：我们打算建立一个版本的瑟斯顿定理的所有postsingularly有限超越整函数，并制定必要的组合结构分类一大类postsingularly有限整函数。

In the 1980's, Bill Thurston developed a unified theory of $3$-manifolds, surface automorphisms, and iterated rational maps; the resulting theorems all rely on an iteration procedure in a finite-dimensional Teichmüller space.Specifically for rational maps, his theorem is of fundamental importance in the description of the iterated maps in terms of symbolic dynamics: in the structurally important case of postcritically finite maps, the stiffness of the invariant complex structure makes it possible to extract finitely many combinatorial invariants that suffice to distinguish different maps, and the classification of the holomorphic maps is based on Thurston's theorem. In each particular case, finding and classifying the appropriate combinatorial invariants is a theorem in its own right. Such invariants have been found and classified for iterated polynomials (Hubbard trees and critical portraits), but not for general rational maps (with the notable exception of rational maps that arise as Newton maps of polynomials). For about 30 years, serious attempts have been made to extend Thurston's theorem from rational to transcendental maps. The only available "transcendental" extension of Thurston's theorem, by Hubbard, Shishikura, and the proposer, covers only the simple family of exponential maps. However, it lays the foundations for extensions to more general transcendental maps.All the fundamental concepts for describing the dynamics of polynomials (dynamic rays, Hubbard trees, spiders, critical portraits, etc), are not easily available for transcendental functions, but significant progress has been made recently in considerable generality, for instance, in proving the existence of dynamic rays (aka "hairs") for many families of maps. This project aims to extend Thurston's fundamental theorem, and the resulting successful classification of postcritically finite polynomials, to the transcendental world: we intend to establish a version of Thurston's theorem for all postsingularly finite transcendental entire functions, and to develop the necessary combinatorial structure to classify a large class of postsingularly finite entire functions.