Some Stabilities of Group Automorphisms

  • Akihiko Morimoto
Part of the Progress in Mathematics book series (PM, volume 14)


Let φ:M→M be a homeomorphism of a metric space (M,d) with distance function d. A (double) sequence {xi}i∈M of points xi ∈ M is called a δ-pseudo-orbit of φ iff d(φ(xi),xi+1) ≤ δ for every i∈Z, where δ>0 is a constant (cf. [2]). Given ε>0, a δ-pseudo-orbit {xi} is called to be ε-traced by a point y∈M iff d(φi (y),xi)≤ε for every i≤Z. We shall call φ stochastically stable, iff for any ε>0 there exists δ>0 such that every δ-pseudo-orbit of φ can be ε-traced by some point y∈M.


Riemannian Manifold Group Automorphism Stochastic Stability Linear Automorphism Jordan Canonical Form 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D.V. Anosov, “Geodesic flows on closed Riemannian manifolds of negative curvature,” Proc. Steklov Inst. Math. 90 (1967).Google Scholar
  2. [2]
    R. Bowen, “ω-limit sets for Axiom A di ffeomorphisms,” J. Diff. Eq. 18 (1975), 333–339.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    J. Franks, “Necessary conditions for stability of diffeomorphisms,” Trans. Amer. Math. Soc. 158 (1971), 301–308.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    G. Hochschild, The Structure of Lie Groups, Holden-Day, 1965.zbMATHGoogle Scholar
  5. [5]
    K. Kato, “Stochastic stability of Anosov diffeomorphisms,” Nagoya Math. J. 69 (1978), 121–129.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Z. Nitecki, “On semi-stability for diffeomorphisms,” Inv. Math. 14 (1971), 83–122.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Z. Nitecki-M. Shub, “Filtrat ions, decompositions and explosions,” Amer. J. Math. 97 (1976), 1029–1047.MathSciNetzbMATHGoogle Scholar
  8. [8]
    J. Robbin, “Topological conjugacies and structural stabilities for discrete dynamical systems,” Bull. Amer. Math. Soc. 78 (1972), 923–952.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J. Robbin, “A structural stability theorem,” Ann. of Math. 94 (1971), 447–493.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    S. Smale, “Differentiable dynamical systems,” Bull. Amer. Math. Soc. 73 (1967), 747–817.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    P. Walters, “Anosov diffeomorphisms are topologically stable,” Topology 9 (1970), 71–78.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    P. Walters, “On the pseudo-orbit tracing property and its relationship to stability,” Springer Lect. Notes. 668 (1979), 231–244.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Akihiko Morimoto
    • 1
  1. 1.Nagoya UniversityChigusaku, Nagoya 464Japan

Personalised recommendations