Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Periodic points and normal families


Authors: Detlef Bargmann and Walter Bergweiler
Journal: Proc. Amer. Math. Soc. 129 (2001), 2881-2888
MSC (2000): Primary 30D05, 30D45, 37F10
DOI: https://doi.org/10.1090/S0002-9939-01-05864-6
Published electronically: February 9, 2001
MathSciNet review: 1840089
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

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.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 30D05, 30D45, 37F10


Additional Information

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

Walter Bergweiler
Affiliation: Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany
Email: bergweiler@math.uni-kiel.de

DOI: https://doi.org/10.1090/S0002-9939-01-05864-6
Received by editor(s): September 13, 1999
Received by editor(s) in revised form: January 31, 2000
Published electronically: February 9, 2001
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society