Infinite-dimensional Thurston theory and transcendental dynamics III: entire functions with escaping singular orbits in the degenerate case

TL;DR

This paper extends infinite-dimensional Thurston theory to classify certain transcendental entire functions, specifically those with escaping singular orbits in degenerate cases, using an iteration on Teichmüller space.

Contribution

It introduces a novel classification method for transcendental entire functions with degenerate escaping singular orbits via infinite-dimensional Thurston theory.

Findings

Classification of functions with escaping singular orbits

Application of infinite-dimensional Teichmüller space iteration

Extension of Thurston's classical theory to transcendental dynamics

Abstract

We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. We focus on the case where the escape is degenerate in the sense that points from different singular orbits are arbitrarily close to each other. As in the general case we employ an iteration procedure on an infinite-dimensional Teichm\"uller space, analogously to the Thurston's classical Topological Characterization of Rational Functions, but for an infinite set of marked points.