Fatou-Julia theory in differentiable dynamics

  • Pei-Chu Hu
  • Chung-Chun Yang
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 4)


Recently, Fornaess and Sibony [9] studied the Fatou-Julia theory of complex dynamics on the complex projective spaces P m . Zhang and Ren ([25]) studied the problems on C m and asked when a Julia set is nonempty? In the paper [12], we introduced our joint work on Fatou-Julia theory in high dimensional spaces. In particular, we proved an existence theorem of fixed points for holomorphic self-mappings on C m . We also obtained a sufficient and necessary condition of attractive fixed points and a sufficient condition of repulsive fixed points for holomorphic self-mappings on complex manifolds, and characterized attractive and repulsive cycles of continuous self-mappings on topological spaces.


Filtration Manifold Betti 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Anosov, D. V., Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR 145 (4) (1962), 707–709.MathSciNetGoogle Scholar
  2. Anosov, D. V., English transi.: Soy. Math. Dokl. 3 (1962), 1068–1070.MATHGoogle Scholar
  3. [2]
    Anosov, D. V., Geodesic flows on closed Riemannian manifolds of negative curvature, Tr. Mat. Inst. Steklova 90. English transl.: Proc. Steklov Inst. Math. 90(1969).Google Scholar
  4. [3]
    Arnold, V. I., Sur une propriété topologique des applications globalement canoniques e à mécanique classique, C. R. Acad. Sci. Paris 261 (1965), 3719–3722.MathSciNetGoogle Scholar
  5. [4]
    Arnold, V. I., Mathematical methods in classical mechanics, Appendix 9, Springer 1978.Google Scholar
  6. [5]
    Bedford, E. & Smillie, J., Polynomial diffeomorphisms of C2. II: stable manifolds and recurrence, J. of Amer. Math. Soc. 4 (1991), 657–679.MathSciNetMATHGoogle Scholar
  7. [6]
    Douady, A., Julia sets of polynomials, Preprint, 1994.Google Scholar
  8. [7]
    Floer, A., Morse theory for Lagrangian intersections, J. Diff. Geom. 28 (1988), 513–547.MathSciNetMATHGoogle Scholar
  9. [8]
    Floer, A., Symplectic fixed points and holomorphic spheres, preprint, Courant Institute, New York University, 1988.Google Scholar
  10. [9]
    Fornaess, J. E. & Sibony, N., Complex dynamics in higher dimension. I, S. M. F. Astérisque 222 (1994), 201–231.MathSciNetGoogle Scholar
  11. [10]
    Hofer, H., Ljusternik-Schnirelman-theory for Lagrangian intersections, Ann. Henri Poincaré-analyse nonlinéaire 5 (1988), 465–499.MathSciNetMATHGoogle Scholar
  12. [11]
    Hu, P. C. & Yang, C. C., Differentiable Dynamics and Complex Dynamics, manuscript.Google Scholar
  13. [12]
    Hu, P. C. Si Yang, C. C., Dynamics in high dimensional spaces, Preprint.Google Scholar
  14. [13]
    Hurewicz, W. & Wallman, H., Dimension theory, Princeton Univ. Press, Princeton, N. J., 1942.Google Scholar
  15. [14]
    Katok, A. & Hasselblatt, B., Introduction to the modern theory of dynamical systems, Encyclopedia of Math. Si its Appl. 54(1995), Cambridge Univ. Press.Google Scholar
  16. [15]
    Milnor, J., Morse theory, Princeton Univ. Press, Princeton, N.J., 1963.MATHGoogle Scholar
  17. [16]
    Pesin, Ya. B. & Sinai, Ya. G., Hyperbolicity and stochasticity of dynamical systems, Mathematical Physics Reviews, Gordon and Breach Press, Harwood Acad. Publ., USA, pp. 53–115, 1981.Google Scholar
  18. [17]
    Pollicott, M., Lectures on ergodic theory and Pesin theory on compact manifolds, London Math. Soc. Lecture Note series 180(1993), Cambridge Univ. Press.Google Scholar
  19. [18]
    Qiu, W. Y., Ren, F. Y. & Yin, Y. C., Iterate for small random perturbations of rational functions and polynomials, Preprint.Google Scholar
  20. [19]
    Shub, M., Global stability of dynamical systems, Springer-Verlag, 1987.Google Scholar
  21. [20]
    Smale, S., Dynamical systems on n-dimensional manifolds, Differential Equations and Dynamical Systems, Proc. Int. Puerto Rico 1965, 483–486.Google Scholar
  22. [21]
    Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.MathSciNetMATHCrossRefGoogle Scholar
  23. [22]
    Wu, H., Normal families of holomorphic mappings, Acta Math. 119(1967), 193–233.MathSciNetMATHCrossRefGoogle Scholar
  24. [23]
    Yin, Y. C., Discontinuity of Julia sets for polynomials, Acta Math. Sinica 38 (1995), 99–102.MathSciNetMATHGoogle Scholar
  25. [24]
    Zehnder, E., The Arnold conjecture for fixed points of symplectic mappings and periodic solutions of Hamiltonian systems, Proceedings of the International Congress of Mathematicians, Berkeley 1986, pp. 1237–1246.Google Scholar
  26. [25]
    Zhang, W. J. & Ren, F. Y., Iterations of holomorphic self-maps of C N, J. of Fudan University(Natural Science) 33 (1994), 452 - 462.MathSciNetMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Pei-Chu Hu
    • 1
    • 2
  • Chung-Chun Yang
    • 1
    • 2
  1. 1.Department of MathematicsShandong UniversityJinanP.R. China
  2. 2.Department of MathematicsThe Hong Kong University of Science and TechnologyKowloon, Hong KongChina

Personalised recommendations