Absence of line fields and Mañé's theorem for nonrecurrent transcendental functions

Let $f:\mathbb{C}\to\hat{\mathbb{C}}$ be a transcendental meromorphic function. Suppose that the finite part $\mathcal{P}(f)\cap \mathbb{C}$ 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 Mañé (1993) about the branching of iterated preimages of disks, and a theorem of McMullen (1994) regarding the absence of invariant line fields for ``measurably transitive'' functions. Both our theorems extend results previously obtained by Graczyk, Kotus and Świątek (2004).