Absence of Line Fields and Mañé’s Theorem for Non-recurrent Transcendental Functions
نویسندگان
چکیده
Let f : C → Ĉ be a transcendental meromorphic function. Suppose that the finite part P(f)∩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 [Ma] about the branching of iterated preimages of disks, and a theorem of McMullen [McM2, Theorem 3.17] regarding absence of invariant line fields for “measurably transitive” functions. Both our theorems extend results previously obtained by Graczyk, Kotus and Świa֒tek [GKŚ].
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