manuscripta mathematica

, Volume 93, Issue 1, pp 109–128 | Cite as

Stability of orbit spaces of endomorphisms

  • Pei-Dong Liu


In this paper it is first proved that, for a hyperbolic set of aC 1 (non-invertible) endomorphism of a compact manifold, the dynamical structure of its orbit space (inverse limit space) is stable underC 1-small perturbations and is semi-stable underC 0-small perturbations. It is then proved that if an Axiom A endomorphism satisfies no-cycle condition then its orbit space is Θ-stable andR-stable underC 1-small perturbations and is semi-Θ-stable and semi-R-stable underC 0-small perturbations.

1991 Mathematics Subject Classification

58F10 58F15 58F11 

Key words and phrases

Orbit space hyperbolic set Axiom A 


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  1. [1]
    W. M. Boothby,An Introduction to Differential Manifolds and Riemannian Geometry, Academic Press, INC, 1986.Google Scholar
  2. [2]
    G. J. Butler, G. Pianigiani,Periodic points and chaotic functions in the unit interval, Bull. Austral. Math. Soc.18 (1978), 255–265.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Z.-P. Chen, L.-F. He, S.-L. Yang,Orbit-shift Θ-stability, Scientia Sinica (Ser. A)31 (1988), 512–520.MathSciNetGoogle Scholar
  4. [4]
    M. Hirsch, J. Palis, C. Pugh, M. Shub,Neighbourhoods of hyperbolic sets, Invent. math.9 (1970), 121–134.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P.-D. Liu, M. Qian,Smooth Ergodic Theory of Random Dynamical Systems, Lec. Not. Math.1606, Springer, 1995.Google Scholar
  6. [6]
    R. Mañé,A proof of the C 1-stability conjecture, IHES Publ. Math.66 (1988), 161–210.zbMATHGoogle Scholar
  7. [7]
    R. Mañé, C. Pugh,Stability of endomorphisms, Lec. Not. Math.468, Springer-Verlag, 1974, 175–184.CrossRefGoogle Scholar
  8. [8]
    Z. Nitecki,On semi-stability for diffeomorphisms, Invent. math.14 (1971), 83–122.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Palis,On the C 1 Θ-stability conjecture, IHES Publ. Math.66 (1988), 211–215.zbMATHMathSciNetGoogle Scholar
  10. [10]
    S. Yu. Pilyugin,The Space of Dynamical Systems with the C 0-Topology, Lec. Not. Math.1571, Springer-Verlag, 1994.Google Scholar
  11. [11]
    F. Przytycki,Anosov endomorphisms, Studia Math.58 (1976), 249–285.zbMATHMathSciNetGoogle Scholar
  12. [12]
    F. Przytycki,Θ-stability and structural stability of endomorphisms, Studia Math.60 (1977), 61–77.zbMATHMathSciNetGoogle Scholar
  13. [13]
    C. Pugh, M. Shub,Ergodic attractors, Trans. Amer. Math. Soc.1 (1989), 1–54.CrossRefMathSciNetGoogle Scholar
  14. [14]
    M. Qian, Z.-S. Zhang,Ergodic theory of Axiom A endomorphisms, Ergod. Th. Dynam. Sys.1 (1995), 161–174.MathSciNetGoogle Scholar
  15. [15]
    V. A. Rokhlin,Exact endomorphisms of a Lebesgue space, AMS Transl. (2)39 (1964), 1–36.zbMATHGoogle Scholar
  16. [16]
    M. Shub,Endomorphisms of compact differentiable manifolds, Amer. J. Math.91 (1969), 171–199.CrossRefMathSciNetGoogle Scholar
  17. [17]
    P. Walters,An Introduction to Ergodic Theory, New York, Springer, 1982.zbMATHGoogle Scholar
  18. [18]
    S.-L. Yang,“More-to-one” hyperbolic endomorphisms and hyperbolic sets, Acta Mathematica Sinica3 (1986), 420–427 (in Chinese).Google Scholar
  19. [19]
    S.-L. Yang,Orbit-shift structural stability of hyperbolic covering endomorphisms, Acta Mathematica Sinica5 (1986), 590–594 (in Chinese).Google Scholar
  20. [20]
    Z.-S. Zhang,Expanding invariant sets of endomorphisms, Science in China (Ser. A)5 (1984), 408–416 (in Chinese).Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Pei-Dong Liu
    • 1
  1. 1.Department of Mathematics and Institute of MathematicsPeking UniversityBeijingP. R. China

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