Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Quasiregular mappings of polynomial type in $ \mathbb{R}^{2}$

Authors: Alastair Fletcher and Dan Goodman
Journal: Conform. Geom. Dyn. 14 (2010), 322-336
MSC (2010): Primary 30C65; Secondary 30D05, 37F10, 37F45
Published electronically: November 23, 2010
MathSciNet review: 2738532
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Complex dynamics deals with the iteration of holomorphic functions. As is well known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in $ \mathbb{R}^{2}$ which are a composition of a quadratic polynomial and an affine stretch.

References [Enhancements On Off] (What's this?)

  • 1. A. F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991. MR 1128089 (92j:30026)
  • 2. W. Bergweiler, Karpinska's paradox in dimension three, Duke. Math. J. 154 (2010), 599-630.
  • 3. W. Bergweiler, A. Eremenko, Dynamics of a higher dimensional analog of the trigonometric functions, preprint.
  • 4. W. Bergweiler, A. Fletcher, J. Langley, J. Meyer, The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc., 137, no. 2, 641-651, 2009. MR 2448586 (2010f:30045)
  • 5. A.Douady, J.H.Hubbard, Itération des polynômes quadratiques complexes [Iteration of complex quadratic polynomials], C. R. Acad. Sci. Paris Sér. I Math., 294, no. 3, 123-126 (1982). MR 651802 (83m:58046)
  • 6. A.Fletcher, V.Markovic, Quasiconformal maps and Teichmüller theory, OUP, 2007.
  • 7. A.Fletcher, D.A.Nicks, Quasiregular dynamics on the n-sphere, to appear in Ergodic Theory and Dynamical Systems.
  • 8. A. Hinkkanen, G. Martin, V. Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand., 95, no. 1, 80-100, 2004. MR 2091483 (2005f:37094)
  • 9. T. Iwaniec, G. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn., 21, no. 2, 241-254, 1996. MR 1404085 (97i:30032)
  • 10. T. Iwaniec, G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, Oxford University Press, New York, 2001. MR 1859913 (2003c:30001)
  • 11. J. Milnor, Dynamics in one complex variable, Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • 12. S.Morosawa, Y.Nishimura, M.Taniguchi and T.Ueda, Holomorphic dynamics, CUP, 2000. MR 1747010 (2002c:37064)
  • 13. S. Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26, Springer, 1993. MR 1238941 (95g:30026)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30C65, 30D05, 37F10, 37F45

Retrieve articles in all journals with MSC (2010): 30C65, 30D05, 37F10, 37F45

Additional Information

Alastair Fletcher
Affiliation: Institute of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom

Dan Goodman
Affiliation: Equipe Audition, Département d’Etudes Cognitives, Ecole Normale Supérieure, 29 Rue d’Ulm 75230, Paris, Cedex 05, France

Keywords: Quasiregular dynamics
Received by editor(s): June 1, 2010
Published electronically: November 23, 2010
Additional Notes: The first author is supported by EPSRC grant EP/G050120/1.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society