Proceedings of the American Mathematical Society

Author(s): Detlef Bargmann; Walter Bergweiler
Journal: Proc. Amer. Math. Soc. 129 (2001), 2881-2888.
MSC (2000): Primary 30D05, 30D45, 37F10
Posted: February 9, 2001
Retrieve article in: PDF
This article is available free of charge

Let $\mathcal{F}$ be the family of all functions which are holomorphic in some domain and do not have periodic points of some period greater than one there. It is shown that $\mathcal{F}$ is quasinormal, and the sequences in $\mathcal{F}$ which do not have convergent subsequences are characterized. The method also yields a new proof of the result that transcendental entire functions have infinitely many periodic points of all periods greater than one.

1.: L. V. Ahlfors, Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157-194, and Collected Papers, Birkhäuser, Boston, Basel, Stuttgart, 1982, Vol. I, pp. 214-251. MR 84k:01066a
2.: I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), 615-622. MR 30:230
3.: W. Bergweiler, Periodic points of entire functions: proof of a conjecture of Baker, Complex Variables Theory Appl. 17 (1991), 57-72. MR 92m:30048
4.: -, Periodische Punkte bei der Iteration ganzer Funktionen. Habilitationsschrift, Rheinisch-Westfälische Techn. Hochsch., Aachen 1991.
5.: -, A new proof of the Ahlfors five islands theorem, J. Analyse Math. 76 (1998), 337-347. MR 99m:30032
6.: L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, New York, Berlin, Heidelberg, 1993. MR 94h:30033
7.: C.-T. Chuang, Normal Families of Meromorphic Functions, World Scientific, Singapore, 1993.
8.: A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287-343. MR 87f:58083
9.: M. Essén and S. Wu, Fix-points and a normal family of analytic functions, Complex Variables Theory Appl. 37 (1998), 171-178. MR 99m:30069
10.: -, Repulsive fixpoints of analytic functions with applications to complex dynamics, J. London Math. Soc. (2) 62 (2000), 139-148.
11.: P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47 (1926), 337-360.
12.: W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. MR 29:1337
13.: -, Research Problems in Function Theory, Athlone Press, London, 1967. MR 36:359
14.: P. Montel, Leçons sur les familles normales des fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927.
15.: R. Nevanlinna, Eindeutige analytische Funktionen, Springer, Berlin, Göttingen, Heidelberg, 1953. MR 15:208c
16.: P. C. Rosenbloom, L'itération des fonctions entières, C. R. Acad. Sci. Paris 227 (1948), 382-383. MR 10:187a
17.: J. L. Schiff, Normal Families, Springer, New York, Berlin, Heidelberg, 1993. MR 94f:30046
18.: L. Yang, Some recent results and problems in the theory of value-distribution, in Proceedings of the Symposium on Value Distribution Theory in Several Complex Variables, (W. Stoll, ed.), Univ. of Notre Dame Press, Notre Dame Math. Lect. 12 (1992), 157-171. MR 94i:30029

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30D05, 30D45, 37F10

Detlef Bargmann
Affiliation: Mathematisches Seminar, Christian--Albrechts--Universität zu Kiel, Ludewig--Meyn--Str. 4, D--24098 Kiel, Germany
Email: bargmann@math.uni-kiel.de

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS