Topological classification of linear hyperbolic cocycles

  • Nguyen Dinh Cong


In this paper linear hyperbolic cocycles are classified by the relation of topological conjugacy. Roughly speaking, two linear cocycles are conjugate if there exists a homeomorphism which maps their trajectories into each other. The problem of classification of discrete-time deterministic hyperbolic dynamical systems was investigated by Robbin (1972). He proved that there exist 4d classes ofd-dimensional deterministic discrete hyperbolic dynamical systems. We obtain a criterion for topological conjugacy of two linear hyperbolic cocycles and show that the number of classes depends crucially on the ergodic properties of the metric dynamical system over which they are defined. Our result is a generalization of the deterministic theorem of Robbin.

Key words

Linear hyperbolic cocycle random homeomorphism Oseledets splitting orientation dimension coboundary 

AMS Subject Classification

S8F15 28D05 58F19 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akcoglu, M. A., and Chacon, R. V. (1965). Generalized eigenvalues of automorphisms.Proc. Am. Math. Soc. 16, 676–680.Google Scholar
  2. Arnold, L. (1994).Random Dynamical Systems. Preliminary version 2, Bremen.Google Scholar
  3. Arnold, L., and Crauel, H. (1991). Random dynamical systems. In Arnold, L., Crauel, H., and Eckmann, J.-P. (Eds) (1991).Lyapunov exponents, Oberwolfach 1990, Lecture Notes in Mathematics, Volume 1486, pp. 1–22, Springer-Verlag, Berlin.Google Scholar
  4. Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G. (1982).Ergodic Theory, Springer-Verlag, New York.Google Scholar
  5. Deimling, K. (1985).Nonlinear Functional Analysis, Springer-Berlag, Berlin.Google Scholar
  6. Dold, A. (1972).Lectures on Algebraic Topology, Springer-Verlag, Berlin.Google Scholar
  7. Gantmacher, F. R. (1977).The Theory of Matrices, Vol. 1, Chelsea, New York.Google Scholar
  8. Gol'dsheid, I. Ya., and Margulis, G. A. (1989). Lyapunov indices of products of random matrices.Russ. Math. Surv. 44 (5), 11–71.Google Scholar
  9. Halmos, P. R. (1956).Lectures on Ergodic Theory, Chelsea, New York.Google Scholar
  10. Irwin, M. C. (1980).Smooth Dynamical Systems, Academic Press, London.Google Scholar
  11. Kato, T. (1976).Perturbation Theory for Linear Operators, Springer-Verlag, Berlin.Google Scholar
  12. Kirillov, A. A. (1967). Dynamical systems, factors and representations of groups,Russ. Math. Surv. 22, 63–75.Google Scholar
  13. Knill, O. (1991). The upper Lyapunov exponent ofSl(2,R) cocycles: Discontinuity and the problem of positivity. In Arnold, L., Crauel, H., and Eckmann, J.-P. (Eds.),Lyapunov Exponents, Oberwolfach 1990, Lecture Notes in Mathematics, Vol. 1486, Springer-Verlag, Berlin, pp. 86–97.Google Scholar
  14. Knill, O. (1992). Positive Lyapunov exponents for a dense set of bounded measurableSl(2,R) cocycles.Ergodic Theory Dynam. Syst. 12 (2), 319–331.Google Scholar
  15. Moore, C. C., and Schmidt, K. (1986). Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson,Proc. London Math. Soc. 40, 443–475.Google Scholar
  16. Oseledets, V. I. (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc. 19, 197–231.Google Scholar
  17. Pears, A. R. (1975).Dimension Theory of General Spaces, Cambridge University Press, Cambridge.Google Scholar
  18. Robbin, J. W. (1972). Topological conjugacy and structural stability for discrete dynamical systems.Bull. Am. Math. Soc. 78 (6), 923–952.Google Scholar
  19. Schmidt, K. (1990).Algebraic Ideas in Ergodic Theory. Regional Conference Series in Mathematics, Number 76, Am. Math. Soc., Providence, RI.Google Scholar
  20. Virtser, A. D. (1979). On products of random matrices and operators.Theory Prob. App. 24, 367–377.Google Scholar
  21. Wanner, T. (1992). A Hartman-Grobman Theorem for Discrete Random Dynamical Systems, Institut für Mathematik, Universität Augsburg, Report Nr. 269.Google Scholar
  22. Wanner, T. (1994). Linearization of random dynamical systems. In John, C., Kirchgraber, U., and Walther, H. O. (Eds.),Dynamics Reported, Vol. 4, Springer, Berlin/Heidelberg/New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Nguyen Dinh Cong
    • 1
    • 2
  1. 1.Institut für Dynamische SystemeUniversität BremenBremenGermany
  2. 2.Hanoi Institute of MathematicsHanoiVietnam

Personalised recommendations