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

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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).