Ledrappier, F.; Young, L.-S. Stability of Lyapunov exponents. (English) Zbl 0759.58026 Ergodic Theory Dyn. Syst. 11, No. 3, 469-484 (1991). 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)]. Reviewer: G.R.Goodson (Towson) Cited in 1 ReviewCited in 8 Documents MSC: 37A99 Ergodic theory 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 15B51 Stochastic matrices Keywords:stability; compact metric space; Lyapunov exponents; stochastic flows Citations:Zbl 0641.60066 PDFBibTeX XMLCite \textit{F. Ledrappier} and \textit{L. S. Young}, Ergodic Theory Dyn. Syst. 11, No. 3, 469--484 (1991; Zbl 0759.58026) Full Text: DOI References: [1] Kifer, J. d’Analyse Math. 47 pp 111– (1986) [2] Ikeda, Stochastic Differential Equations and Diffusion Processes (1981) [3] DOI: 10.1007/BF00535004 · Zbl 0529.60025 · doi:10.1007/BF00535004 [4] DOI: 10.1007/BF02760620 · Zbl 0528.60028 · doi:10.1007/BF02760620 [5] Young, Ergod. Th. & Dynam. Sys. none pp 627– (1986) [6] Kifer, Progress in Probability and Statistics (1986) [7] DOI: 10.1007/BF01223205 · Zbl 0659.60092 · doi:10.1007/BF01223205 [8] DOI: 10.1007/BF00356103 · Zbl 0638.60054 · doi:10.1007/BF00356103 [9] Kunita, Stochastic differential equations and stochastic flow of diffeomorphisms 1097 (1984) · Zbl 0554.60066 [10] Kifer, Ergod. Th. & Dynam. Sys. 2 pp 367– (1982) [11] DOI: 10.1137/1121063 · Zbl 0367.60035 · doi:10.1137/1121063 [12] Ruelle, Publ. Math. IHES 50 pp 27– (1979) · Zbl 0426.58014 · doi:10.1007/BF02684768 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.