Analysis and Mathematical Physics

, Volume 4, Issue 1–2, pp 83–98 | Cite as

The fast escaping set for quasiregular mappings

  • Walter Bergweiler
  • David Drasin
  • Alastair Fletcher


The fast escaping set of a transcendental entire function is the set of all points which tend to infinity under iteration as fast as possible compatible with the growth of the function. We study the analogous set for quasiregular mappings in higher dimensions and show, among other things, that various equivalent definitions of the fast escaping set for transcendental entire functions in the plane also coincide for quasiregular mappings. We also exhibit a class of quasiregular mappings for which the fast escaping set has the structure of a spider’s web.

Mathematics Subject Classification (2000)

Primary 37F10 Secondary 30C65 30D05 


  1. 1.
    Bell, H.: On fixed point properties of plane continua. Trans. Amer. Math. Soc. 128, 539–548 (1967)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bergweiler, W.: Fixed points of composite entire and quasiregular maps. Ann. Acad. Sci. Fenn. Math. 31, 523–540 (2006)MATHMathSciNetGoogle Scholar
  3. 3.
    Bergweiler, W.: Karpinska’s paradox in dimension 3. Duke Math. J. 154, 599–630 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bergweiler, W.: Iteration of quasiregular mappings. Comput. Methods Funct. Theory 10, 455–481 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bergweiler, W.: Fatou–Julia theory for non-uniformly quasiregular maps. Ergodic Theory Dynam. Syst. 33, 1–23 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bergweiler, W., Fletcher, A., Langley, J.K., Meyer, J.: The escaping set of a quasiregular mapping. Proc. Amer. Math. Soc. 137, 641–651 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bergweiler, W., Hinkkanen, A.: On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126, 565–574 (1999)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bergweiler, W., Nicks, D.A.: Foundations for an iteration theory of entire quasiregular maps. to appear in Isr. J. Math.Google Scholar
  9. 9.
    Bergweiler, W., Rippon, P.J., Stallard, G.M.: Dynamics of meromorphic functions with direct or logarithmic singularities. Proc. Lond. Math. Soc. 97, 368–400 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Blokh, A., Oversteegen, L.: A fixed point theorem for branched covering maps of the plane. Fundam. Math. 206, 77–111 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Drasin, D., Pankka, P.: Sharpness of Rickman’s Picard theorem in all dimensions. arXiv:1304.6998
  12. 12.
    Drasin, D., Sastry, S.: Periodic quasiregular mappings of finite order. Rev. Mat. Iberoamericana 19, 755–766 (2003)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Eremenko, A.E.: On the iteration of entire functions. Dynamical systems and ergodic theory, Banach Center Publications 23. Polish Scientific Publishers, Warsaw (1989)Google Scholar
  14. 14.
    Fletcher, A., Nicks, D.A.: Quasiregular dynamics on the \(n\)-sphere. Ergod. Theory Dynam. Syst. 31, 23–31 (2011)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Macintyre, A.J.: Wiman’s method and the ‘flat regions’ of integral functions. Q. J. Math. Oxford Ser. 9, 81–88 (1938)CrossRefGoogle Scholar
  16. 16.
    Mihaljevic-Brandt, H., Peter, J.: Poincaré functions with spiders’ webs. Proc. Amer. Math. Soc. 140, 3193–3205 (2012)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Miniowitz, R.: Normal families of quasimeromorphic mappings. Proc. Amer. Math. Soc. 84, 35–43 (1982)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Newman, M.H.A.: Elements of the topology of plane sets of points. Cambridge University Press, New York (1961)Google Scholar
  19. 19.
    Osborne, J.W.: The structure of spider’s web fast escaping sets. Bull. Lond. Math. Soc. 44, 503–519 (2012)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Rickman, S.: On the number of omitted values of entire quasiregular mappings. J. Anal. Math. 37, 100–117 (1980)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Rickman, S.: The analogue of Picard’s theorem for quasiregular mappings in dimension three. Acta Math. 154, 195–242 (1985)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rickman, S.: Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26. Springer, New York (1993)Google Scholar
  23. 23.
    Rippon, P., Stallard, G.: On questions of Fatou and Eremenko. Proc. Amer. Math. Soc. 133, 1119–1126 (2005)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Rippon, P., Stallard, G.: Fast escaping points of entire functions. Proc. Lond. Math. Soc. 105, 787–820 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Rippon, P., Stallard, G.: A sharp growth condition for a fast escaping spider’s web. Adv. Math. 244, 337–353 (2013)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Siebert, H.: Fixed points and normal families of quasiregular mappings. J. Anal. Math. 98, 145–168 (2006)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Walter Bergweiler
    • 1
  • David Drasin
    • 2
  • Alastair Fletcher
    • 3
  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu Kiel KielGermany
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA
  3. 3.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

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