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


Authors: Lasse Rempe and Sebastian van Strien
Journal: Trans. Amer. Math. Soc. 363 (2011), 203-228
MSC (2010): Primary 37F10; Secondary 30D05, 37D25, 37F15, 37F35
DOI: https://doi.org/10.1090/S0002-9947-2010-05125-6
Published electronically: August 26, 2010
MathSciNet review: 2719679
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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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Additional Information

Lasse Rempe
Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom
Email: l.rempe@liverpool.ac.uk

Sebastian van Strien
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: strien@maths.warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2010-05125-6
Received by editor(s): October 8, 2008
Published electronically: August 26, 2010
Additional Notes: This research was supported by EPSRC grant EP/E017886/1.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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