Science China Mathematics

, Volume 53, Issue 3, pp 593–596 | Cite as

Normal families and fixed points of iterates

Articles

Abstract

Let \( \mathcal{F} \) be a family of holomorphic functions and suppose that there exists ɛ > 0 such that if f\( \mathcal{F} \), then |(f 2)′(ξ)| ⩽ 4 − ɛ for all fixed points ξ of the second iterate f 2. We show that then \( \mathcal{F} \) is normal. This is deduced from a result which says that if p is a polynomial of degree at least 2, then p 2 has a fixed point ξ such that |(p 2)′(ξ)| ⩾ 4. The results are motivated by a problem posed by Yang Lo.

Keywords

fixed point multiplier normality iteration periodic point 

MSC(2000)

Primary 30D45 Secondary 30D05, 37F10 

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References

  1. 1.
    Bargmann D, Bergweiler W. Periodic points and normal families. Proc Amer Math Soc, 2001, 129: 2881–2888MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bergweiler W. Quasinormal families and periodic points. In: Complex analysis and dynamical systems II, 55–63, Contemp. Math. vol. 382. Providence, RI: Amer Math Soc, 2005Google Scholar
  3. 3.
    Bergweiler W. Bloch’s principle. Comput Methods Funct Theory, 2006, 6: 77–108MATHMathSciNetGoogle Scholar
  4. 4.
    Chang J, Fang M. Normal families and fixed points. J Anal Math, 2005, 95: 389–395MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chang J, Fang M, Zalcman L. Normality and attracting fixed points. Bull London Math Soc, 2008, 40: 777–788MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Erëmenko A È, Levin G M. Periodic points of polynomials (in Russian). Ukrain Mat Zh, 1989, 41: 1467–1471; Translation in Ukrainian Math J, 1990, 41: 1258–1262MATHGoogle Scholar
  7. 7.
    Essén M, Wu S J. Fix-points and a normal family of analytic functions. Complex Variables Theory Appl, 1998, 37: 171–178MATHMathSciNetGoogle Scholar
  8. 8.
    Essén M, Wu S J. Repulsive fix-points of analytic functions with applications to complex dynamics. J London Math Soc, 2000, 62: 139–149MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pang X C. Shared values and normal families. Analysis, 2002, 22: 175–182MATHGoogle Scholar
  10. 10.
    Pang X C, Zalcman L. Normal families and shared values. Bull London Math Soc, 2000, 32: 325–331MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Siebert H. Fixed points and normal families of quasiregular mappings. J Anal Math, 2006, 98: 145–168MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Wang S G, Wu S J. Fix-points of meromorphic functions and quasinormal families (in Chinese). Acta Math Sinica Chin Ser, 2002, 45: 545–550MATHMathSciNetGoogle Scholar
  13. 13.
    Xu Y. On Montel’s theorem and Yang’s problem. J Math Anal Appl, 2005, 305: 743–751MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Yang L. Some recent results and problems in the theory of value-distribution. In: Proceedings of the symposium on value distribution theory in several complex variables, 157–171, Notre Dame Math. Lectures 12. Notre Dame: University of Notre Dame Press, 1992Google Scholar
  15. 15.
    Zalcman L. A heuristic principle in complex function theory. Amer Math Monthly, 1975, 82: 813–817MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Zalcman L. Normal families: new perspectives. Bull Amer Math Soc, 1998, 35: 215–230MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Mathematisches Seminar der Christian-Albrechts-Universität zu KielKielGermany

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