On convergence of Thurston's iteration for entire functions with an infinite set of marked points

Abstract

The goal of this note is to generalize Thurston's Topological Characterization of Rational Functions to the setting when both the covering degree and the set of marked points are infinite. A relevant class of branched coverings are transcendental entire functions with finitely many singular values whose orbits escape to (or, more generally, accumulate ``near'') $\infty$. Given a branched covering $f$ mimicking such post-singular behaviour, one wants to decide whether it is Thurston equivalent to an entire function. The answer is positive for a big class of entire function and generic escaping singular orbits. As in the Thurston's theorem, the problem reduces to the study of the pull-back map $\sigma$ defined on the corresponding Teichm\"uller space. But, unlike in the rational case, the space is infinite-dimensional and the branching structure near $\infty$ (which is essential singularity) is much more subtle depending on the family of functions under consideration. A general approach is possible for entire functions defined by asymptotic area property introduced in the article. Roughly, it implies that asymptotic tracts fill all space near $\infty$ even when their range shrinks. The main result provides a sufficient condition for existence in the Teichm\"uller space of a $\sigma$-invariant subset which looks like a finite-dimensional compact: forgetting marked infinite tails of every orbit yields a compact set, while forgetting a long enough initial part of every orbit yields a small perturbation of the identity homeomorphism. The statement remains valid even in case of non-escaping unbounded singular orbits and allows to deduce existence of a fixed point of $\sigma$ in many relevant cases.