Dynamics of transcendental functions with escaping singular orbits and infinite-dimensional Teichmüller theory

动力学问题理论的基本问题之一是确定哪些系统是等价的，如何区分不同的系统，以及如何对不同的动力学可能性进行分类。这个一般问题应该在迭代先验映射的背景下进行研究。该研究项目的目标是研究有限类型（即具有有限多个临界值和渐近值）的某些迭代整体超越函数的动力学，其特性是所有临界值和渐近值在迭代下收敛到无穷大。临界值和渐近值的轨道形成一个离散集合 P，仅在无穷远处累积。对于有限类型整个函数的适当族，这些点的组合和渐近应该允许我们给出各个整个函数的分类。一个可能的扩展涉及那些有限类型超越函数，其中所有临界值和渐近值要么收敛到无穷大（如之前），要么是周期或前周期的（或者可能收敛到吸引循环）。这项研究的重要工具是（无限维）Teichmüller 空间理论，该空间是根据黎曼球中 P 的补集建模的。为了实现这一目标，有必要将瑟斯顿定理（有时也称为“复杂动力学基本定理”）从后批判有限有理映射（使用有限维泰希米勒理论）扩展到无限维上下文，以及从有理映射到超越映射（即从有限到无限映射度的情况）。

One of the fundamental questions in the theory of dynamical questions is to determine which systems are equivalent, how different systems can be distinguished, and how the different dynamical possibilities can be classified. This general question shall be investigated in the context of iterated transcendental mappings. Goal of this research project is the investigation of the dynamics of certain iterated entire transcendental functions of finite type (that is, with finitely many critical and asymptotical values) with the property that all critical and asymptotic values converge to infinity under iteration. The orbits of critical and asymptotic values form a discrete set P that accumulates only at infinity. For appropriate families of entire functions of finite type the combinatorics and asymptotics of these points should allow us go give a classification of the respective entire functions.A possible extension concers those finite type transcendental functions for which all critical and asymptotic values either converge to infinity (as before) or are periodic or preperiodic (or possibly converge to attracting cycles). Important tool for this investigation will be the theory of (infinite-dimensional) Teichmüller spaces that are modeled after the complement of P in the Riemann sphere. To accomplish this, it will be necessary to extend Thurston's theorem (that is sometimes also called the "fundamental theorem of complex dynamics") from postcritically finite rational maps (which uses finite dimensional Teichmüller theory) to an infinite dimensional context, and also from rational to transcendental maps (that is, from the case of finite to infinite mapping degrees).