Hyperbolic sets and shift automorhpisms

  • Charles C. Conley
III. Nonlinear Differential Equations
Part of the Lecture Notes in Physics book series (LNP, volume 38)


An elementary discussion of hyperbolic invariant sets is presented and a proof of the theorem that “near any nonperiodic chain recurrent point there lies an embedded shift automorphism” is indicated.


Periodic Orbit Periodic Point Stable Manifold Vertical Strip Homoclinic Point 


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  1. [1]
    Selgrade, J., Isolated invariant sets for flows on vector bundles (to appear).Google Scholar
  2. [2]
    Moser, J., Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics; Annals of Mathematics Studies 77, Princeton University Press 1973.Google Scholar
  3. [3]
    Smale, S., Diffeomorphisms with many periodic points. Differential and Combinatorial Topology (edited by S. S. Cairns), Princeton University Press (1965) 63–80.Google Scholar
  4. [4]
    Bowen, R., w limit sets for axiom A diffeomorphisms (to appear in Jour. Diff. Equa).Google Scholar
  5. [5]
    Newhouse, S., “Hyperbolic limit sets,” Trans. Amer. Math. Soc. 167 (1972) 125–150.Google Scholar
  6. [6]
    Pugh, C. C., and Shub, M., “The Ω-stability theorem for flows,” Inventiones Math. 11 (1970) 150–158.Google Scholar
  7. [7]
    Shub, M., “Stability and Genericity for Diffeomorphisms,” in Dynamical Systems (ed. M. Peixoto) Proccedings of Symposium on Dynamical Systems, Salvador, Bahia, Brazil (1971) Academic Press.Google Scholar
  8. [8]
    Smale, S., “The Ω-Stability Theorem,” in Global Analysis, Proc. Symp. Pure Math. 14 (Providence: A.M.S. 1970) 289–299.Google Scholar
  9. [9]
    Palis, J., “On Morse-Smale Dynamical Systems,” Topology 4 (1969) 385–404.Google Scholar
  10. [10]
    Anosov, D. V., Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. 90 (1967).Google Scholar
  11. [11]
    Bowen, R., Topological entropy and Axiom A, Proc. Symp. Pure Math. 14 (1970) 23–41.Google Scholar
  12. [12]
    Bowen, R., Periodic points and measures for axiom A diffeomorphisms, Trans. AMS 154 (1971) 377–397.Google Scholar
  13. [13]
    Hirsch, M., Palis, J., Pugh, C., and Shub, M., Neighborhoods of hyperbolic sets, Inventiones Math. 9 (1970) 121–134.Google Scholar
  14. [14]
    Hirsch, M., and Pugh, C., Stable manifolds and hyperbolic sets, Proc. Symp. Pure Math. 14 (1970) 133–163.Google Scholar
  15. [15]
    Smale, S., Differentiable dynamical systems, Bull. AMS 73 (1967) 747–817.Google Scholar
  16. [16]
    Walters, P., Anosov diffeomorphisms are topologically stable, Topology 9 (1970) 71–78.Google Scholar
  17. [17]
    Guckenheimer, J., “Absolutely Ω-stable diffeomorphisms,” Topology 11 (1972), 195–197.Google Scholar
  18. [18]
    Hirsch, M., Palis, J., Pugh, C., and Shub, M., “Neighborhoods of hyperbolic sets,” Inventiones Math. 9 (1970) 121–134.Google Scholar
  19. [19]
    Mane, R., “Persistent manifolds are normally hyperbolic, to appear.Google Scholar
  20. [20]
    Moser, J., “On a theorem of Anosov,” J. Diff. Equa. 5 (1969) 411–440.Google Scholar
  21. [21]
    Fenichel, N., “Persistence and smoothness of invariant manifolds for flows,” Indiana Univ. Math. J. 21 (1971) 193–226.Google Scholar
  22. [22]
    Franks, J., “Necessary conditions for stability of diffeomorphisms,” Trans. AMS 158 (1972) 301–308.Google Scholar
  23. [23]
    Franks, J., “Differentiably Ω-stable diffeomorphisms,” Topology 11 (1972) 107–113.Google Scholar
  24. [24]
    Franks, J., “Time dependent stable diffeomorphisms,” Inventiones Math., 24 (1974), pp. 163–172.Google Scholar
  25. [25]
    Robbin, J., “A structural stability theorem,” Annals Math. 94 (1971) 447–493.Google Scholar
  26. [26]
    Robinson, R.C., “Structural stability of C1 flows,” to appear Proc. Conf. Applic. Topology and Dynamical Systems, Univ. of Warwick, 1974.Google Scholar
  27. [27]
    Sacker, R., “A perturbation theorem for invariant manifolds and Hőlder continuity,” J. Math. Mech. 18 (1969) 705–762.Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Charles C. Conley
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadison

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