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

Rempe, Lasse; van Strien, Sebastian

doi:10.1090/S0002-9947-2010-05125-6

Absence of line fields and Mañé’s theorem for nonrecurrent transcendental functions
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by Lasse Rempe and Sebastian van Strien

Trans. Amer. Math. Soc. 363 (2011), 203-228

DOI: https://doi.org/10.1090/S0002-9947-2010-05125-6

Published electronically: August 26, 2010

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Abstract:

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).

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