Skip to main content
SpringerLink
Log in

The fast escaping set for quasiregular mappings

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bell, H.: On fixed point properties of plane continua. Trans. Amer. Math. Soc. 128, 539–548 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bergweiler, W.: Fixed points of composite entire and quasiregular maps. Ann. Acad. Sci. Fenn. Math. 31, 523–540 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Bergweiler, W.: Karpinska’s paradox in dimension 3. Duke Math. J. 154, 599–630 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bergweiler, W.: Iteration of quasiregular mappings. Comput. Methods Funct. Theory 10, 455–481 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bergweiler, W.: Fatou–Julia theory for non-uniformly quasiregular maps. Ergodic Theory Dynam. Syst. 33, 1–23 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bergweiler, W., Hinkkanen, A.: On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126, 565–574 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bergweiler, W., Nicks, D.A.: Foundations for an iteration theory of entire quasiregular maps. to appear in Isr. J. Math.

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Blokh, A., Oversteegen, L.: A fixed point theorem for branched covering maps of the plane. Fundam. Math. 206, 77–111 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Drasin, D., Pankka, P.: Sharpness of Rickman’s Picard theorem in all dimensions. arXiv:1304.6998

  12. Drasin, D., Sastry, S.: Periodic quasiregular mappings of finite order. Rev. Mat. Iberoamericana 19, 755–766 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. Fletcher, A., Nicks, D.A.: Quasiregular dynamics on the \(n\)-sphere. Ergod. Theory Dynam. Syst. 31, 23–31 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Macintyre, A.J.: Wiman’s method and the ‘flat regions’ of integral functions. Q. J. Math. Oxford Ser. 9, 81–88 (1938)

    Article  Google Scholar 

  16. Mihaljevic-Brandt, H., Peter, J.: Poincaré functions with spiders’ webs. Proc. Amer. Math. Soc. 140, 3193–3205 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Miniowitz, R.: Normal families of quasimeromorphic mappings. Proc. Amer. Math. Soc. 84, 35–43 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  18. Newman, M.H.A.: Elements of the topology of plane sets of points. Cambridge University Press, New York (1961)

  19. Osborne, J.W.: The structure of spider’s web fast escaping sets. Bull. Lond. Math. Soc. 44, 503–519 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rickman, S.: On the number of omitted values of entire quasiregular mappings. J. Anal. Math. 37, 100–117 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rickman, S.: The analogue of Picard’s theorem for quasiregular mappings in dimension three. Acta Math. 154, 195–242 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rickman, S.: Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26. Springer, New York (1993)

    Google Scholar 

  23. Rippon, P., Stallard, G.: On questions of Fatou and Eremenko. Proc. Amer. Math. Soc. 133, 1119–1126 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rippon, P., Stallard, G.: Fast escaping points of entire functions. Proc. Lond. Math. Soc. 105, 787–820 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rippon, P., Stallard, G.: A sharp growth condition for a fast escaping spider’s web. Adv. Math. 244, 337–353 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Siebert, H.: Fixed points and normal families of quasiregular mappings. J. Anal. Math. 98, 145–168 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 , Kiel, Germany

    Walter Bergweiler

  2. Department of Mathematics, Purdue University, West Lafayette, IN , 47907, USA

    David Drasin

  3. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL , 60115-2888, USA

    Alastair Fletcher

Authors
  1. Walter Bergweiler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Drasin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alastair Fletcher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Walter Bergweiler.

Additional information

Dedicated to Larry Zalcman on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bergweiler, W., Drasin, D. & Fletcher, A. The fast escaping set for quasiregular mappings. Anal.Math.Phys. 4, 83–98 (2014). https://doi.org/10.1007/s13324-014-0078-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-014-0078-9

Mathematics Subject Classification (2000)

Search

Navigation