Abstract
LetF be a family of mappingsK-quasiregular in some domainG. We show that if for eachf∈F, there existsk>1 such that thek-th iteratef k off has no fixed point, thenF is normal. Moreover, we examine to what extent this result holds if we consider only repelling fixed points, rather than fixed points in general. We also prove thatF is quasinormal, ifF contains only quasiregular mappings that do not have periodic points of some period greater than one inG. This implies that a quasiregular mappingf:ℝ n with an essential singularity in ∞ has infinitely many periodic points of any period greater than one. These results generalize results of M. Essén, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Bargmann and W. Bergweiler,Periodic points and normal families, Proc. Amer. Math. Soc.129 (2001), 2881–2888.
W. Bergweiler,Periodic points of entire functions: proof of a conjecture of Baker, Complex Variables Theory Appl.17 (1991), 57–72.
W. Bergweiler,A new proof of the Ahlfors five islands theorem, J. Analyse Math.76 (1998), 337–347.
K. Deimling,Nonlinear Functional Analysis, Springer, Berlin-Heidelberg-New York-Tokyo, 1985.
M. Essén and S. Wu,Fixpoints and a normal family of analytic functions, Complex Variables Theory Appl.37 (1998), 171–178.
M. Essén and S. Wu,Repulsive fixpoints of analytic functions with applications to complex dynamics, J. London Math. Soc. (2)62 (2000), 139–148.
A. Hinkkanen and G. Martin,Attractors in quasiregular semigroups, inProceedings of the XVI Nevanlinna Colloquium (I. Laine and O. Martio, eds.), de Gruyter, Berlin-New York, 1996, pp. 135–141.
A. Hinkkanen, G. Martin and V. Mayer,Local dynamics of uniformly quasiregular mappings, Math. Scand.95 (2004), 80–100.
W. Hurewicz and H. Wallman,Dimension Theory, Princeton Univ. Press, Princeton, NJ, 1948.
T. Iwaniec and G. Martin,Geometric Function Theory and Non-linear Analysis, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2001.
O. Martio, S. Rickman and J. Väisälä,Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math.488 (1971).
O. Martio and U. Srebro,Periodic quasimeromorphic mappings, J. Analyse Math.28 (1975), 20–40.
R. Miniowitz,Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc.84 (1982), 35–43.
Yu. G. Reshetnyak,Space mappings with bounded distortion, Sibirk. Mat. Zh.8 (1967), 629–659.
S. Rickman,On the number of omitted values of entire quasiregular mappings, J. Analyse Math.37 (1980), 100–117.
S. Rickman,Quasiregular Mappings, Springer, Berlin, 1993.
H. Siebert,Fixpunkte und normale Familien quasiregulärer Abbildungen, Dissertation, CAU Kiel, 2004.
J. Väisälä,Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math.392 (1966), 1–10.
S. Wang and S. Wu,Fixpoints of meromorphic functions and quasinormal families, Acta Math. Sin.45 (2002), 545–550.
L. Zalcman,A heuristic principle in complex function theory, Amer. Math. Monthly82 (1975), 813–817.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Siebert, H. Fixed points and normal families of quasiregular mappings. J. Anal. Math. 98, 145–168 (2006). https://doi.org/10.1007/BF02790273
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02790273