Skip to main content Accessibility help

Fatou–Julia theory for non-uniformly quasiregular maps



Many results of the Fatou–Julia iteration theory of rational functions extend to uniformly quasiregular maps in higher dimensions. We obtain results of this type for certain classes of quasiregular maps which are not uniformly quasiregular.



Hide All
[1]Ahlfors, L. V.. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973.
[2]Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics, 91). Springer, New York, 1991.
[3]Bergweiler, W.. Iteration of quasiregular mappings. Comput. Methods Funct. Theory 10 (2010), 455481.
[4]Bergweiler, W.. Karpińska’s paradox in dimension 3. Duke Math. J. 154 (2010), 599630.
[5]Bergweiler, W. and Eremenko, A.. Dynamics of a higher dimensional analogue of the trigonometric functions. Ann. Acad. Sci. Fenn. Math. 36 (2011), 165175.
[6]Bergweiler, W., Fletcher, A., Langley, J. and Meyer, J.. The escaping set of a quasiregular mapping. Proc. Amer. Math. Soc. 137 (2009), 641651.
[7]Eremënko, A. È.. On the iteration of entire functions. Dynamical Systems and Ergodic Theory (Banach Center Publications, 23). Polish Scientific Publishers, Warsaw, 1989, pp. 339345.
[8]Fletcher, A. and Nicks, D. A.. Quasiregular dynamics on the n-sphere. Ergod. Th. & Dynam. Sys. 31 (2011), 2331.
[9]Fletcher, A. and Nicks, D. A.. Julia sets of uniformly quasiregular mappings are uniformly perfect. Math. Proc. Cambridge Philos. Soc., doi:10.1017/S0305004111000478.
[10]Garber, V.. On the iteration of rational functions. Math. Proc. Cambridge Philos. Soc. 84 (1978), 497505.
[11]Gehring, F. W.. A remark on domains quasiconformally equivalent to a ball. Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 147155.
[12]Hinkkanen, A., Martin, G. J. and Mayer, V.. Local dynamics of uniformly quasiregular mappings. Math. Scand. 95 (2004), 80100.
[13]Iwaniec, T. and Martin, G. J.. Geometric Function Theory and Non-linear Analysis (Oxford Mathematical Monographs). Oxford University Press, New York, 2001.
[14]Martin, G. J.. Branch sets of uniformly quasiregular maps. Conform. Geom. Dyn. 1 (1997), 2427.
[15]Martio, O.. A capacity inequality for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I No. 474 (1970) 18 pp.
[16]Mattila, P. and Rickman, S.. Averages of the counting function of a quasiregular mapping. Acta Math. 143 (1979), 273305.
[17]Mayer, V.. Uniformly quasiregular mappings of Lattès type. Conform. Geom. Dyn. 1 (1997), 104111.
[18]Milnor, J.. Dynamics in one Complex Variable, 3rd edn(Annals of Mathematics Studies 160). Princeton University Press, Princeton, NJ, 2006.
[19]Miniowitz, R.. Normal families of quasimeromorphic mappings. Proc. Amer. Math. Soc. 84 (1982), 3543.
[20]Przytycki, F. and Urbañski, M.. Conformal Fractals: Ergodic Theory Methods (London Mathematical Society Lecture Note Series, 371). Cambridge University Press, Cambridge, 2010.
[21]Reshetnyak, Yu. G.. Space Mappings with Bounded Distortion (Translations of Mathematical Monographs, 73). American Mathematical Society, Providence, RI, 1989.
[22]Rickman, S.. On the number of omitted values of entire quasiregular mappings. J. Anal. Math. 37 (1980), 100117.
[23]Rickman, S.. The analogue of Picard’s theorem for quasiregular mappings in dimension three. Acta Math. 154 (1985), 195242.
[24]Rickman, S.. Quasiregular Mappings (Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 26). Springer, Berlin, 1993.
[25]Sarvas, J.. Symmetrization of condensers in n-space. Ann. Acad. Sci. Fenn. Ser. A I No. 522 (1972) 44 pp.
[26]Siebert, H.. Fixpunkte und normale Familien quasiregulärer Abbildungen. Dissertation, University of Kiel, (2004);
[27]Srebro, U.. Quasiregular mappings. Advances in Complex Function Theory (Lecture Notes in Mathematics, 505). Springer, Berlin, 1976, pp. 148163.
[28]Steinmetz, N.. Rational Iteration (De Gruyter Studies in Mathematics, 16). Walter de Gruyter & Co, Berlin, 1993.
[29]Sun, D. and Yang, L.. Quasirational dynamical systems (Chinese). Chinese Ann. Math. Ser. A 20 (1999), 673684.
[30]Sun, D. and Yang, L.. Quasirational dynamic system. Chinese Science Bull. 45 (2000), 12771279.
[31]Sun, D. and Yang, L.. Iteration of quasi-rational mapping. Prog. Nat. Sci. (English Ed.) 11 (2001), 1625.
[32]Vuorinen, M.. Some inequalities for the moduli of curve families. Michigan Math. J. 30 (1983), 369380.
[33]Vuorinen, M.. Conformal Geometry and Quasiregular Mappings (Lecture Notes in Mathematics, 1319). Springer, Berlin, 1988.
[34]Wallin, H.. Metrical characterization of conformal capacity zero. J. Math. Anal. Appl. 58 (1977), 298311.
[35]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.
[36]Zalcman, L.. A heuristic principle in complex function theory. Amer. Math. Monthly 82 (1975), 813817.

Fatou–Julia theory for non-uniformly quasiregular maps



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed