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Structural stability of linear random dynamical systems

Published online by Cambridge University Press:  14 October 2010

Nguyen Dinh Cong
Nguyen Dinh Cong
Affiliation:
Institut für Dynamische Systeme, Universität Bremen, Postfach 330 440, 28334 Bremen, Germany
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Abstract

In this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1207 - 1220
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[Arn] Arnold, L.. Random Dynamical Systems. Preliminary version 2, Bremen, 1994.Google Scholar
[AC] Arnold, L. and Crauel, H.. Random dynamical systems. Lyapunov Exponents, Oberwolfach 1990 (Lecture Notes in Mathematics 1486). Eds Arnold, L., Crauel, H. and Eckmann, J.-P.. Springer, Berlin, 1991, pp. 1–22.Google Scholar
[Box] Boxler, P.. A stochastic version of center manifold theory. Probab. Th. Rel. Fields 83 (1989), 509–545.CrossRefGoogle Scholar
[C] Cong, Nguyen Dinh. Topological classification of linear hyperbolic cocycles. J. Dyn. Differential Equations. To appear.Google Scholar
[Gun] Gundlach, V. M.. Random homoclinic orbits. Random & Computational Dynamics 3 (1995), 1–33.Google Scholar
[Irw] Irwin, M. C.. Smooth Dynamical Systems. Academic, London, 1980.Google Scholar
[Joh] Johnson, R. A.. Remarks on linear differential systems with measurable coefficients. Proc. Amer. Math. Soc. 100 (1987), 491–504.CrossRefGoogle Scholar
[Kn] Knill, O.. The upper Lyapunov exponent of Sl(2, R) cocycles: discontinuity and the problem of positivity. Lyapunov Exponents, Oberwolfach 1990 (Lecture Notes in Mathematics I486). Eds Arnold, L., Crauel, H. and Eckmann, J.-P.. Springer, Berlin, 1991, pp. 86–97.Google Scholar
[Os] Oseledets, V. I.. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[Pal] Palmer, K. J.. Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynamics Reported 1 (1988), 265–306.CrossRefGoogle Scholar
[Rob] Robbin, J. W.. Topological conjugacy and structural stability for discrete dynamical systems. Bull. Amer. Math. Soc. 78 (1972), 923–952.CrossRefGoogle Scholar