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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
##### Keywords:
index of exponential dichotomy
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