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.
Similar content being viewed by others
References
I. N. Baker, The domains of normality of an entire function, Annales Academiae Scientiarium Fennicae. Mathematica 1 (1975), 277–283.
W. Bergweiler, Iteration of meromorphic functions, Bulletin of the American Mathematical Society 29 (1993), 151–188.
W. Bergweiler, Fixed points of composite entire and quasiregular maps, Annales Academiae Scientiarium Fennicae. Mathematica 31 (2006), 523–540.
W. Bergweiler, Iteration of quasiregular mappings, Computational Methods and Function Theory 10 (2010), 455–481.
W. Bergweiler, Karpińska’s paradox in dimension3, Duke Mathematical Journal 154 (2010), 599–630.
W. Bergweiler, On the set where the iterates of an entire function are bounded, Proceedings of the American Mathematical Society 140 (2012), 847–853.
W. Bergweiler, Fatou-Julia theory for non-uniformly quasiregular maps, Ergodic Theory and Dynamical Systems 33 (2013), 1–23.
W. Bergweiler and A. Eremenko, Dynamics of a higher dimensional analogue of the trigonometric functions, Annales Academiae Scientarium Fennicae. Mathematica 36 (2011), 165–175.
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.
R. L. Devaney, Complex exponential dynamics, in Handbook of Dynamical Systems, Vol.3, Elsevier, Amsterdam, 2010, pp. 125–224.
R. L. Devaney and M. Krych, Dynamics of exp(z), Ergodic Theory and Dynamical Systems 4 (1984), 35–52.
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.
A. Eremenko and I. Ostrovskii, On the’ pits effect’ of Littlewood and Offord, Bulletin of the London Mathematical Society 39 (2007), 929–939.
P. Fatou, Sur les équations fonctionelles, Bulletin de la Société Mathématique de France 47 (1919), 161–271; 48 (1920), 33–94, 208–314.
P. Fatou, Sur l’itération des fonctions transcendantes entières, Acta Mathematica 47 (1926), 337–360.
A. Fletcher and D. A. Nicks, Quasiregular dynamics on the n-sphere, Ergodic Theory and Dynamical Systems 31 (2011), 23–31.
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.
A. Fletcher and D. A. Nicks, Chaotic dynamics of a quasiregular sine mapping, Journal of Difference Equations and Applications 19 (2013), 1353–1360.
T. Iwaniec and G. J. Martin, Geometric Function Theory and Non-linear Analysis, Oxford Mathematical Monographs, Oxford University Press, New York, 2001.
G. Julia, Sur l’itération des fonctions rationelles, Journal de Mathématiques Pures et Appliquées 4 (1918), 47–245.
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.
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.
P. Mattila and S. Rickman, Averages of the counting function of a quasiregular mapping, Acta Mathematica 143 (1979), 273–305.
C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Transactions of the American Mathematical Society 300 (1987), 329–342.
J. Milnor, Dynamics in One Complex Variable, third edition, Annals of Mathematics Studies, Vol. 160, Princeton University Press, Princeton, NJ, 2006.
R. Miniowitz, Normal families of quasimeromorphic mappings, Proceedings of the American Mathematical Society 84 (1982), 35–43.
D. A. Nicks, Wandering domains in quasiregular dynamics, Proceedings of the American Mathematical Society 141 (2013), 1385–1392.
L. Rempe, Topological dynamics of exponential maps on their escaping sets, Ergodic Theory and Dynamical Systems 26 (2006), 1939–1975.
Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, Vol. 73, American Mathematical Society, Providence, RI, 1989.
S. Rickman, On the number of omitted values of entire quasiregular mappings, Journal d’Analyse Mathématique 37 (1980), 100–117.
S. Rickman, The analogue of Picard’s theorem for quasiregular mappings in dimension three, Acta Mathematica 154 (1985), 195–242.
S. Rickman, Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 26, Springer-Verlag, Berlin, 1993.
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.
D. Schleicher, Attracting dynamics of exponential maps, Annales Academiae Scientarium Fennicae. Mathematica 28 (2003), 3–34.
H. Siebert, Fixpunkte und normale Familien quasiregulärer Abbildungen, Dissertation, University of Kiel, 2004; http://e-diss.uni-kiel.de/diss_1260.
H. Siebert, Fixed points and normal families of quasiregular mappings, Journal d’Analyse Mathématique 98 (2006), 145–168.
N. Steinmetz, Rational Iteration, De Gruyter Studies in Mathematics, Vol. 16, Walter de Gruyter & Co., Berlin 1993.
D. Sun and L. Yang, Quasirational dynamical systems, (Chinese) Chinese Annals of Mathematics. Series A 20 (1999), 673–684.
D. Sun and L. Yang, Quasirational dynamic system, Chinese Science Bulletin 45 (2000), 1277–1279.
D. Sun and L. Yang, Iteration of quasi-rational mapping, Progress in Natural Science. English Edition 11 (2001), 16–25.
H. Wallin, Metrical characterization of conformal capacity zero, Journal of Mathematical Analysis and Applications 58 (1977), 298–311.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is supported by the Deutsche Forschungsgemeinschaft, Be 1508/7-2 and the ESF Networking Programme HCAA
Rights and permissions
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-014-1081-4