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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)

Abstract

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.

Keywords

Lyapunov Exponent Local Coordinate System Morse Index Borel Probability Measure Measure Preserve Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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

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