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

动力学问题理论中的一个基本问题是确定哪些系统是等效的，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。这个一般问题将在迭代超越映射的背景下进行研究。本研究的目的是研究一类有限型（即具有有限个临界值和渐近值）的迭代整体超越函数的动力学性质，这些函数在迭代下具有所有的临界值和渐近值收敛于无穷。临界值和渐近值的轨道形成一个只在无穷远处累积的离散集P。对于适当的有限型完整函数族，这些点的组合学和渐近性应该允许我们给出相应的完整函数的分类。一个可能的扩展涉及那些有限型超越函数，其所有临界值和渐近值要么收敛于无穷（如前所述），要么是周期的或前周期的（或可能收敛于吸引环）。这项研究的重要工具将是（无限维）teichm<e:1>空间理论，该理论是在黎曼球中P的补后建模的。为了实现这一点，有必要将瑟斯顿定理（有时也称为“复杂动力学的基本定理”）从后批判的有限有理映射（使用有限维的teichm<s:1> ller理论）扩展到无限维的背景，并从有理映射扩展到超越映射（即从有限映射度扩展到无限映射度）。

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).