Absence of line fields and Mane's theorem for non-recurrent transcendental functions

TL;DR

This paper extends classical results about rational functions to transcendental meromorphic functions, proving the absence of invariant line fields under certain dynamical conditions, thus advancing understanding of their complex dynamics.

Contribution

It generalizes Mane's and McMullen's theorems to transcendental functions, establishing conditions for the absence of invariant line fields.

Findings

No invariant line fields on Julia sets under specified conditions

Generalization of rational function theorems to transcendental setting

Extension of previous results by Graczyk, Kotus, and Swiatek

Abstract

Let f be a transcendental meromorphic function. Suppose that the finite part of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded. Then we show that f supports no invariant line fields on its Julia set. We prove this by generalizing two results about rational functions to the transcendental setting: a theorem of Mane about the branching of iterated preimages of disks, and a theorem of McMullen regarding absence of invariant line fields for "measurably transitive" functions. Both our theorems extend results previously obtained by Graczyk, Kotus and Swiatek.