Skip to main content

Foundations for an iteration theory of entire quasiregular maps

Abstract

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type, the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.

This is a preview of subscription content, access via your institution.

References

  1. I. N. Baker, The domains of normality of an entire function, Annales Academiae Scientiarium Fennicae. Mathematica 1 (1975), 277–283.

    Article  MATH  Google Scholar 

  2. W. Bergweiler, Iteration of meromorphic functions, Bulletin of the American Mathematical Society 29 (1993), 151–188.

    MathSciNet  Article  MATH  Google Scholar 

  3. W. Bergweiler, Fixed points of composite entire and quasiregular maps, Annales Academiae Scientiarium Fennicae. Mathematica 31 (2006), 523–540.

    MathSciNet  MATH  Google Scholar 

  4. W. Bergweiler, Iteration of quasiregular mappings, Computational Methods and Function Theory 10 (2010), 455–481.

    MathSciNet  Article  MATH  Google Scholar 

  5. W. Bergweiler, Karpińska’s paradox in dimension3, Duke Mathematical Journal 154 (2010), 599–630.

    MathSciNet  Article  MATH  Google Scholar 

  6. W. Bergweiler, On the set where the iterates of an entire function are bounded, Proceedings of the American Mathematical Society 140 (2012), 847–853.

    MathSciNet  Article  MATH  Google Scholar 

  7. W. Bergweiler, Fatou-Julia theory for non-uniformly quasiregular maps, Ergodic Theory and Dynamical Systems 33 (2013), 1–23.

    MathSciNet  Article  MATH  Google Scholar 

  8. W. Bergweiler and A. Eremenko, Dynamics of a higher dimensional analogue of the trigonometric functions, Annales Academiae Scientarium Fennicae. Mathematica 36 (2011), 165–175.

    MathSciNet  Article  MATH  Google Scholar 

  9. W. Bergweiler, A. Fletcher, J. Langley and J. Meyer, The escaping set of a quasiregular mapping, Proceedings of the American Mathematical Society 137 (2009), 641–651.

    MathSciNet  Article  MATH  Google Scholar 

  10. R. L. Devaney, Complex exponential dynamics, in Handbook of Dynamical Systems, Vol.3, Elsevier, Amsterdam, 2010, pp. 125–224.

    Chapter  Google Scholar 

  11. R. L. Devaney and M. Krych, Dynamics of exp(z), Ergodic Theory and Dynamical Systems 4 (1984), 35–52.

    MathSciNet  Article  MATH  Google Scholar 

  12. A. È. Eremënko, On the iteration of entire functions, in Dynamical Systems and Ergodic Theory (Warsaw 1986), Banach Center Publications, Vol. 23, Polish Scientific Publishers, Warsaw, 1989, pp. 339–345.

    Google Scholar 

  13. A. Eremenko and I. Ostrovskii, On the’ pits effect’ of Littlewood and Offord, Bulletin of the London Mathematical Society 39 (2007), 929–939.

    MathSciNet  Article  MATH  Google Scholar 

  14. P. Fatou, Sur les équations fonctionelles, Bulletin de la Société Mathématique de France 47 (1919), 161–271; 48 (1920), 33–94, 208–314.

    MathSciNet  MATH  Google Scholar 

  15. P. Fatou, Sur l’itération des fonctions transcendantes entières, Acta Mathematica 47 (1926), 337–360.

    MathSciNet  Article  MATH  Google Scholar 

  16. A. Fletcher and D. A. Nicks, Quasiregular dynamics on the n-sphere, Ergodic Theory and Dynamical Systems 31 (2011), 23–31.

    MathSciNet  Article  MATH  Google Scholar 

  17. A. Fletcher and D. A. Nicks, Julia sets of uniformly quasiregular mappings are uniformly perfect, Mathematical Proceedings of the Cambridge Philosophical Society 151 (2011), 541–550.

    MathSciNet  Article  MATH  Google Scholar 

  18. A. Fletcher and D. A. Nicks, Chaotic dynamics of a quasiregular sine mapping, Journal of Difference Equations and Applications 19 (2013), 1353–1360.

    MathSciNet  Article  MATH  Google Scholar 

  19. T. Iwaniec and G. J. Martin, Geometric Function Theory and Non-linear Analysis, Oxford Mathematical Monographs, Oxford University Press, New York, 2001.

    Google Scholar 

  20. G. Julia, Sur l’itération des fonctions rationelles, Journal de Mathématiques Pures et Appliquées 4 (1918), 47–245.

    Google Scholar 

  21. B. Karpińska, Hausdorff dimension of the hairs without endpoints for λ exp z, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 328 (1999), 1039–1044.

    MATH  Google Scholar 

  22. J. E. Littlewood and A. C. Offord, On the distribution of zeros and a-values of a random integral function. II, Annals of Mathematics 49 (1948) 885–952; errata 50 (1949), 990–991.

    MathSciNet  Article  Google Scholar 

  23. P. Mattila and S. Rickman, Averages of the counting function of a quasiregular mapping, Acta Mathematica 143 (1979), 273–305.

    MathSciNet  Article  MATH  Google Scholar 

  24. C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Transactions of the American Mathematical Society 300 (1987), 329–342.

    MathSciNet  Article  MATH  Google Scholar 

  25. J. Milnor, Dynamics in One Complex Variable, third edition, Annals of Mathematics Studies, Vol. 160, Princeton University Press, Princeton, NJ, 2006.

    MATH  Google Scholar 

  26. R. Miniowitz, Normal families of quasimeromorphic mappings, Proceedings of the American Mathematical Society 84 (1982), 35–43.

    MathSciNet  Article  MATH  Google Scholar 

  27. D. A. Nicks, Wandering domains in quasiregular dynamics, Proceedings of the American Mathematical Society 141 (2013), 1385–1392.

    MathSciNet  Article  MATH  Google Scholar 

  28. L. Rempe, Topological dynamics of exponential maps on their escaping sets, Ergodic Theory and Dynamical Systems 26 (2006), 1939–1975.

    MathSciNet  Article  MATH  Google Scholar 

  29. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, Vol. 73, American Mathematical Society, Providence, RI, 1989.

    MATH  Google Scholar 

  30. S. Rickman, On the number of omitted values of entire quasiregular mappings, Journal d’Analyse Mathématique 37 (1980), 100–117.

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  32. S. Rickman, Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 26, Springer-Verlag, Berlin, 1993.

    Book  MATH  Google Scholar 

  33. P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proceedings of the American Mathematical Society 127 (1999), 3251–3258.

    MathSciNet  Article  MATH  Google Scholar 

  34. D. Schleicher, Attracting dynamics of exponential maps, Annales Academiae Scientarium Fennicae. Mathematica 28 (2003), 3–34.

    MathSciNet  MATH  Google Scholar 

  35. H. Siebert, Fixpunkte und normale Familien quasiregulärer Abbildungen, Dissertation, University of Kiel, 2004; http://e-diss.uni-kiel.de/diss_1260.

  36. H. Siebert, Fixed points and normal families of quasiregular mappings, Journal d’Analyse Mathématique 98 (2006), 145–168.

    MathSciNet  Article  MATH  Google Scholar 

  37. N. Steinmetz, Rational Iteration, De Gruyter Studies in Mathematics, Vol. 16, Walter de Gruyter & Co., Berlin 1993.

    Book  MATH  Google Scholar 

  38. D. Sun and L. Yang, Quasirational dynamical systems, (Chinese) Chinese Annals of Mathematics. Series A 20 (1999), 673–684.

    MathSciNet  MATH  Google Scholar 

  39. D. Sun and L. Yang, Quasirational dynamic system, Chinese Science Bulletin 45 (2000), 1277–1279.

    Article  Google Scholar 

  40. D. Sun and L. Yang, Iteration of quasi-rational mapping, Progress in Natural Science. English Edition 11 (2001), 16–25.

    MathSciNet  Google Scholar 

  41. H. Wallin, Metrical characterization of conformal capacity zero, Journal of Mathematical Analysis and Applications 58 (1977), 298–311.

    MathSciNet  Article  MATH  Google Scholar 

  42. V. A. Zorich, A theorem of M. A. Lavrent’ev on quasiconformal space maps, Mathematics of the USSR-Sbornik 3 (1967), 389–403; Translation of Matematicheskiı Sbornik. Novaya Seriya 74 (1967), 417–433 (in Russian).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Walter Bergweiler.

Additional information

The first author is supported by the Deutsche Forschungsgemeinschaft, Be 1508/7-2 and the ESF Networking Programme HCAA

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bergweiler, W., Nicks, D.A. Foundations for an iteration theory of entire quasiregular maps. Isr. J. Math. 201, 147–184 (2014). https://doi.org/10.1007/s11856-014-1081-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-014-1081-4

Keywords

  • Entire Function
  • Periodic Point
  • Local Index
  • Transcendental Entire Function
  • Iteration Theory