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arXiv · 0802.0666

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

Lasse Rempe·Sebastian van Strien
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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.

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Lasse Rempe, Sebastian van Strien. 2008-02-05. Absence of line fields and Mane's theorem for non-recurrent transcendental functions. https://doi.org/10.1090/s0002-9947-2010-05125-6

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