Proceedings of the American Mathematical Society

Author(s): Walter Bergweiler; Alastair Fletcher; Jim Langley; Janis Meyer
Journal: Proc. Amer. Math. Soc. 137 (2009), 641-651.
MSC (2000): Primary 30C65, 30C62; Secondary 37F10.
Posted: September 4, 2008
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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

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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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: 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
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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