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On the hyperbolicity of smooth cocycles over flows with invariant ergodic measure. (Russian. English summary) Zbl 0604.58040

The author shows that if \(\Phi(p,t)\) is a smooth h-dimensional cocycle with Lyapunov exponents \(\lambda_ i\neq 0\), \(i=1,...,n\) such that exactly k among them are negative then for any \(\epsilon >0\) there is a number \(\alpha >0\) and a measurable subset \(M_{\epsilon}\) with measure \(>1-\epsilon\) such that for any \(p\in M\) there exists a k-dimensional linear subspace \(L^+(p)\) satisfying \(| \Phi(p,t)z| \leq a | z| e^{-\alpha t}\) for any \(z\in L^+(p)\) and \(t\geq 0\), where \(\lambda =1/2\min_{i}| \lambda_ i|\), and \(| \Phi(p,t)z| \leq \alpha | z| e^{\lambda t}\) for any \(t\leq 0\) and z from the complementary \((n-k)\)-dimensional linear subspace \(L^- (p)\).
Reviewer: Yu.Kifer

MSC:

37D99 Dynamical systems with hyperbolic behavior
34D15 Singular perturbations of ordinary differential equations
37A99 Ergodic theory
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