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.
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Akcoglu, M. A., and Chacon, R. V. (1965). Generalized eigenvalues of automorphisms.Proc. Am. Math. Soc. 16, 676–680.
Arnold, L. (1994).Random Dynamical Systems. Preliminary version 2, Bremen.
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.
Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G. (1982).Ergodic Theory, Springer-Verlag, New York.
Deimling, K. (1985).Nonlinear Functional Analysis, Springer-Berlag, Berlin.
Dold, A. (1972).Lectures on Algebraic Topology, Springer-Verlag, Berlin.
Gantmacher, F. R. (1977).The Theory of Matrices, Vol. 1, Chelsea, New York.
Gol'dsheid, I. Ya., and Margulis, G. A. (1989). Lyapunov indices of products of random matrices.Russ. Math. Surv. 44 (5), 11–71.
Halmos, P. R. (1956).Lectures on Ergodic Theory, Chelsea, New York.
Irwin, M. C. (1980).Smooth Dynamical Systems, Academic Press, London.
Kato, T. (1976).Perturbation Theory for Linear Operators, Springer-Verlag, Berlin.
Kirillov, A. A. (1967). Dynamical systems, factors and representations of groups,Russ. Math. Surv. 22, 63–75.
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.
Knill, O. (1992). Positive Lyapunov exponents for a dense set of bounded measurableSl(2,R) cocycles.Ergodic Theory Dynam. Syst. 12 (2), 319–331.
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.
Oseledets, V. I. (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc. 19, 197–231.
Pears, A. R. (1975).Dimension Theory of General Spaces, Cambridge University Press, Cambridge.
Robbin, J. W. (1972). Topological conjugacy and structural stability for discrete dynamical systems.Bull. Am. Math. Soc. 78 (6), 923–952.
Schmidt, K. (1990).Algebraic Ideas in Ergodic Theory. Regional Conference Series in Mathematics, Number 76, Am. Math. Soc., Providence, RI.
Virtser, A. D. (1979). On products of random matrices and operators.Theory Prob. App. 24, 367–377.
Wanner, T. (1992). A Hartman-Grobman Theorem for Discrete Random Dynamical Systems, Institut für Mathematik, Universität Augsburg, Report Nr. 269.
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.
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Cong, N.D. Topological classification of linear hyperbolic cocycles. J Dyn Diff Equat 8, 427–467 (1996). https://doi.org/10.1007/BF02218762
- Linear hyperbolic cocycle
- random homeomorphism
- Oseledets splitting
AMS Subject Classification