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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.
Published electronically: September 4, 2008
MathSciNet review: 2448586
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Abstract: We show that if the maximum modulus of a quasiregular mapping $ f: \mathbb{R}^N \to \mathbb{R}^N$ grows sufficiently rapidly, then there exists a nonempty escaping set $ I(f)$ consisting of points whose forward orbits under iteration of $ f$ tend to infinity. We also construct a quasiregular mapping for which the closure of $ I(f)$ 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 $ I(f)$ are unbounded and where it is in fact conjectured that all components of $ I(f)$ are unbounded.


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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
Published electronically: 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.