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Stability of Lyapunov exponents. (English) Zbl 0759.58026

From the abstract: “We consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly volume preserving diffeomorphisms.”
Specifically, let \(f: (X,m)\to(X,m)\) be a homeomorphism of a compact metric space which preserves a probability measure \(m\), and let \(A: X\to GL(q,R)\) be a continuous map for some fixed \(q\in\mathbb{Z}^ +\). It is shown that when \(f\) and \(A\) are in the Lipschitz category, Lyapunov exponents are always stable in a well defined sense, provided that certain regularity conditions hold. For example it is required that the distributions of the random matrices at each step have densities that scale properly with \(\varepsilon\) as \(\varepsilon\to 0\). These results apply to stochastic flows preserving a fixed volume.
This paper generalizes results of the second author [Ergodic Theory Dyn. Syst. 6, 627-637 (1986; Zbl 0641.60066)].

MSC:

37A99 Ergodic theory
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
15B51 Stochastic matrices

Citations:

Zbl 0641.60066
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References:

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[2] Ikeda, Stochastic Differential Equations and Diffusion Processes (1981)
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[4] DOI: 10.1007/BF02760620 · Zbl 0528.60028 · doi:10.1007/BF02760620
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