ABSENCE OF LINE FIELDS AND MANE'S THEOREM FOR NONRECURRENT TRANSCENDENTAL FUNCTIONS

Let f : C -> (C) over cap be a transcendental meromorphic function. Suppose that the finite part P(f)boolean AND 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 Mane (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 Swiatek (2004).