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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The escaping set of a quasiregular mapping
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by Walter Bergweiler, Alastair Fletcher, Jim Langley and Janis Meyer PDF
Proc. Amer. Math. Soc. 137 (2009), 641-651 Request permission

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
  • MR Author ID: 35350
  • Email: bergweiler@math.uni-kiel.de
  • Alastair Fletcher
  • Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
  • MR Author ID: 749646
  • Email: alastair.fletcher@nottingham.ac.uk
  • Jim Langley
  • Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
  • MR Author ID: 110110
  • 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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 641-651
  • MSC (2000): Primary 30C65, 30C62; Secondary 37F10
  • DOI: https://doi.org/10.1090/S0002-9939-08-09609-3
  • MathSciNet review: 2448586