Thurston theory in transcendental dynamics

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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.