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The escaping set of a quasiregular mapping
Authors:
Walter Bergweiler, Alastair Fletcher, Jim Langley and Janis Meyer
Journal:
Proc. Amer. Math. Soc. 137 (2009), 641-651
MSC (2000):
Primary 30C65, 30C62; Secondary 37F10.
Posted:
September 4, 2008
MathSciNet review:
2448586
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Abstract: We show that if the maximum modulus of a quasiregular mapping  grows sufficiently rapidly, then there exists a nonempty escaping set  consisting of points whose forward orbits under iteration of  tend to infinity. We also construct a quasiregular mapping for which the closure of  has a bounded component. This stands in contrast to the situation for entire functions in the complex plane, for which all components of the closure of  are unbounded and where it is in fact conjectured that all components of  are unbounded.
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- W. Bergweiler, Fixed points of composite entire and quasiregular maps, Ann. Acad. Sci. Fenn. Math. 31 (2006), 523-540. MR 2248829 (2007g:30043)
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- L. Rempe, On a question of Eremenko concerning escaping components of entire functions, Bull. London Math. Soc. 39 (2007), 661-666. MR 2346947
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Additional Information
Walter Bergweiler
Affiliation:
Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany
Email:
bergweiler@math.uni-kiel.de
Alastair Fletcher
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Email:
alastair.fletcher@nottingham.ac.uk
Jim Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Email:
jkl@maths.nott.ac.uk
Janis Meyer
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Email:
janis.meyer@maths.nottingham.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09609-3
PII:
S 0002-9939(08)09609-3
Received by editor(s):
February 7, 2008
Posted:
September 4, 2008
Additional Notes:
This research was supported by the G.I.F, the German-Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003, and the EU Research Training Network CODY (first author); EPSRC grant RA22AP (second and third authors); the ESF Research Networking Programme HCAA (first and third authors); and DFG grant ME 3198/1-1 (fourth author).
Communicated by:
Mario Bonk
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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