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Absence of line fields and Mane's theorem for nonrecurrent transcendental functions
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Rempe, Lasse and Strien, Sebastian van (2011) Absence of line fields and Mane's theorem for nonrecurrent transcendental functions. Transactions of the American Mathematical Society, Vol.363 (No.1). pp. 203-228. doi:10.1090/S0002-9947-2010-05125-6 ISSN 0002-9947.
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Official URL: http://dx.doi.org/10.1090/S0002-9947-2010-05125-6
Abstract
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).
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Transactions of the American Mathematical Society | ||||
Publisher: | American Mathematical Society | ||||
ISSN: | 0002-9947 | ||||
Official Date: | January 2011 | ||||
Dates: |
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Volume: | Vol.363 | ||||
Number: | No.1 | ||||
Number of Pages: | 26 | ||||
Page Range: | pp. 203-228 | ||||
DOI: | 10.1090/S0002-9947-2010-05125-6 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | EP/E017886/1 |
Data sourced from Thomson Reuters' Web of Knowledge
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