Acta Mathematica Sinica, English Series

, Volume 33, Issue 11, pp 1536–1548 | Cite as

Analytic properties for holomorphic matrix-valued maps in ℂ2×2

  • Chao Fu
  • Ye Zhou Li
  • Xiao Yao


In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on ℂ2×2 and Julia set of one dimensional complex dynamic system.


Picard theorem holomorphic matrix-value functions asymptotic values Julia set 

MR(2010) Subject Classification



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We would like to thank Professor Zhang Guangyuan’s insights, valuable comments and encouragement during the preparation for this paper. And part of this paper has been done during Yao Xiao visits NTNU in Trondheim and Yao Xiao would like to thank Professor John Erik Fornaess’s kindness for providing such a nice opportunity.


  1. [1]
    Ahlfors, L. V.: Complex Analysis, Third Edition, McGraw-Hill, New York, 1979MATHGoogle Scholar
  2. [2]
    Eremenko, A. E., Lyubich, M. Y.: Dynamical properties of some classes of entire functions. Annales de l’institu. Fourier, 42, 989–1020 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Hayman, Walter K.: Meromorphic Functions, Oxford University Press, Oxford, 1964MATHGoogle Scholar
  4. [4]
    Horn, R. A., Charles R. J.: Matrix Analysis, Cambridge University Press, Cambridge, 2012CrossRefGoogle Scholar
  5. [5]
    Milnor, J. W.: Dynamics in One Complex Variable, Princeton University Press, Princeton, 2006MATHGoogle Scholar
  6. [6]
    Misiurewicz, M.: On iterates of e z. Ergodic Theory Dynam. Systems, 1 103–106 (1981)CrossRefMATHGoogle Scholar
  7. [7]
    Rosay, J. P., Walter R.: Holomorphic maps from C n to C n. Trans. Amer. Math. Soc., 310(1), 47–86 (1988)MathSciNetMATHGoogle Scholar
  8. [8]
    Schleicher, D.: Dynamics of Entire Functions, Springer-Verlag, Berlin Heidelberg, 2010CrossRefMATHGoogle Scholar
  9. [9]
    Yang, L.: Value Distribution Theory, Springer-Verlag, New York, 2013Google Scholar
  10. [10]
    Zhan, X. Z: Matrix Inequalities, Springer-Verlag, Berlin-Heidelberg, 2002CrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingP. R. China
  2. 2.Shanghai Center Mathematical SciencesShanghaiP. R. China

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