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Normality and repelling periodic points


Authors: Jianming Chang and Lawrence Zalcman
Journal: Trans. Amer. Math. Soc. 363 (2011), 5721-5744
MSC (2000): Primary 30D45, 30D05, 37F10, 37C25
DOI: https://doi.org/10.1090/S0002-9947-2011-05280-3
Published electronically: June 17, 2011
MathSciNet review: 2817406
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Abstract: Let $ k\ge 3(\ge 2)$ be an integer and $ \mathcal{F}$ be a family of functions meromorphic in a domain $ D\subset\mathbb{C}$, all of whose poles have multiplicity at least 2 (at least 3). If in $ D$ each $ f\in\mathcal{F}$ has neither repelling fixed points nor repelling periodic points of period $ k$, then $ \mathcal{F}$ is a normal family in $ D$. Examples are given to show that the conditions on poles are necessary and sharp.


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  • [1] I. N. Baker, Fix-points of polynomials and rational functions, J. London Math. Soc. (2) 39 (1964), 615-622. MR 0169989 (30:230)
  • [2] D. Bargmann and W. Bergweiler, Periodic points and normal families, Proc. Amer. Math. Soc 129 (2001), 2881-2888. MR 1840089 (2002c:30034)
  • [3] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N. S.) 29 (1993), 151-188. MR 1216719 (94c:30033)
  • [4] W. Bergweiler, Periodische Punkte bei der Iteration ganzer Funktionen, Habilitationsschrift, Rheinisch-Westfälische Techn. Hochsch., Aachen, 1991.
  • [5] W. Bergweiler, An Introduction to Complex Dynamics, Universidade de Coimbra, Coimbra, 1995. MR 1626031 (99c:58133)
  • [6] W. Bergweiler, Ahlfors theory and complex dynamics: Periodic points of entire functions, in ``Complex dynamics and related fields'', RIMS Kokyuroku 1269 (2002), 1-11. MR 1953889
  • [7] W. Bergweiler and N. Terglane, Weakly repulsive fixpoints and the connectivity of wandering domains, Trans. Amer. Math. Soc. 348 (1996), 1-12. MR 1327252 (96e:30055)
  • [8] L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993. MR 1230383 (94h:30033)
  • [9] J. M. Chang, Polynomials without repelling periodic points of given periods, J. Math. Anal. Appl. 324 (2006), 1-13. MR 2262451 (2008a:37047)
  • [10] J. M. Chang, Normality, quasinormality and periodic points, Nagoya Math. J. 195 (2009), 77-95. MR 2552954 (2010h:37092)
  • [11] J. M. Chang and M. L. Fang, Normal families and fixed points, J. Anal. Math. 95 (2005), 389-395. MR 2145570 (2005m:30034)
  • [12] J. M. Chang and M. L. Fang, Repelling periodic points of rational functions of given periods, Sci. China Ser. A 49 (2006), 1165-1174. MR 2284204 (2007k:37055)
  • [13] J. M. Chang, M. L. Fang and L. Zalcman, Normality and attracting fixed points, Bull. London Math. Soc. 40 (2008), 777-788. MR 2439643 (2009g:30038)
  • [14] M. Essén and S. J. Wu, Fix-points and normal families of analytic functions, Complex Variables Theory Appl. 37 (1998), 171-178. MR 1687872 (99m:20069)
  • [15] M. Essén and S. J. Wu, Repulsive fixpoints of analytic functions with application to complex dynamics, J. London Math. Soc. (2) 62 (2000), 139-148. MR 1771857 (2001k:37065)
  • [16] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. MR 0164038 (29:1337)
  • [17] J. Milnor, Dynamics in One Complex Variable, Princeton University Press, Princeton, 2006. MR 2193309 (2006g:37070)
  • [18] X. C. Pang, Shared values and normal families, Analysis 22 (2002), 175-182. MR 1916423 (2003h:30043)
  • [19] X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000), 325-331. MR 1750485 (2001e:30059)
  • [20] J. Schiff, Normal Families, Springer-Verlag, 1993. MR 1211641 (94f:30046)
  • [21] M. Shishikura, On the quasi-conformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), 1-29. MR 892140 (88i:58099)
  • [22] N. Steinmetz, Rational Iteration, Walter de Gruyter, 1993. MR 1224235 (94h:30035)
  • [23] L. Yang, Some recent results and problems in the theory of value distribution, Proceedings Symposium on Value Distribution Theory in Several Complex Variables (Notre Dame, IN, 1990) (W. Stoll, ed.), Univ. Notre Dame Press, 1992, pp. 157-171. MR 1243023 (94i:30029)
  • [24] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. MR 1301781 (95h:30039)
  • [25] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813-817. MR 0379852 (52:757)
  • [26] L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc. (N. S.) 35 (1998), 215-230. MR 1624862 (99g:30048)

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Additional Information

Jianming Chang
Affiliation: Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, People’s Republic of China
Email: jmchang@cslg.edu.cn

Lawrence Zalcman
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
Email: zalcman@macs.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2011-05280-3
Keywords: Meromorphic function, normal family, iterate, fixed point, periodic point.
Received by editor(s): September 6, 2009
Published electronically: June 17, 2011
Additional Notes: The research of the first author was supported by NNSF of China (Grant No. 10871094), NSFU of Jiangsu, China (Grant No. 08KJB110001), Qinglan Project of Jiangsu, China, and the SRF for ROCS, SEM.
The research of the second author was supported by Israel Science Foundation Grant 395/07. This work is part of the European Science Foundation Networking Programme HCAA.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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