## The escaping set of a quasiregular mapping

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- by Walter Bergweiler, Alastair Fletcher, Jim Langley and Janis Meyer
- Proc. Amer. Math. Soc.
**137**(2009), 641-651 - DOI: https://doi.org/10.1090/S0002-9939-08-09609-3
- Published electronically: 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 $I(f)$ 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.## References

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## Bibliographic 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