## 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

### Keywords:

cocycle; Lyapunov exponents
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