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On indices of exponential dichotomy. (Russian) Zbl 0564.34048
Let (E,p,B) be a vector bundle having \({\mathbb{R}}^ n\) as its fibre and the base space B a complete metric space. Let G stand for either the additive group \({\mathbb{R}}\) or \({\mathbb{Z}}\) and let Hom(G,Aut(E,p,B)) denote the set of all homomorphisms of G into the group Aut(E,p,B), the group of automorphisms of the vector bundle. If H is such a homomorphism then for every \(t\in G\) H(t) represents a homeomorphism \(X^ t\) of E onto itself and a homeomorphism \(X^ t\) of B onto itself which satisfy the relation \(pX^ t=x^ tp\) for every \(t\in G\). Endowing the space E with a Riemannian metric the author defines the index of exponential dichotomy \(I_ k(H,b)\) of the homomorphism H at a point \(b\in B\) for \(k\in \{1,2,...,n-1\}\) by the formula \[ I_ k(H,b)=\inf_{{\mathbb{R}}^ k\in G_ k(p^{-1}(b))} \limsup_{t-s\to \infty,(t\in {\mathbb{Z}}^+,s\in {\mathbb{Z}}^+)}\frac{1}{t-s}\ln \{\| X^{t-s}_{X^ s{\mathbb{R}}^ k}\| \| X^{s-t}_{X^ t(R^ k)^ 2}\| \] where \(G_ k(p^-(b))\) stands for the Grassmann manifold of all k-dimensional vector subspaces of the fibre \(p^{-1}(b)\). The homomorphism H is said to have the exponential dichotomy of index k at \(b\in B\) if there exists \({\mathbb{R}}_ 0^ k\in G_ k(p^{-1}(b))\) such that for some orthogonal complement \({\mathbb{R}}^{n-k}\) of \({\mathbb{R}}^ k\) there exist \(\alpha >0\) and \(b>0\) such that for all non-zero vectors \(\xi \in {\mathbb{R}}^{n-k}\), \(\eta \in {\mathbb{R}}^ k\) and for all \(t,s\in {\mathbb{Z}}^+\) with \(t\geq s\) the inequality \(| X^ t\xi | | X^ s\xi |^{-1}\geq \alpha | X^ t\xi | | X^ s\xi |^{-1}\exp (\beta (t- s))\) is satisfied. Among other important results the author shows that the homomorphism H has at the point b the exponential dichotomy of index k if and only if \(I(H,b)<0\).
Reviewer: L.Janos

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
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