Normal bases of a family of endomorphisms of a metrized vector bundle (Q1102430)

Remind that a fundamental system of solutions of the linear system of differential equations \(\dot x=A(t)x\), \(x\in R^ n\), is said to be normal, if the sum of the Lyapunov exponents of the solutions is minimal in the set of all fundamental solutions. A generalization of the concept of a normal fundamental system of solutions is given. Let \((E,p,B)\) be an abstract vector bundle with Riemannian metric (i.e. E,B-sets, \(p: E\to B\), the fibre \(p^{-1}(b)\) has the structure of an n- dimensional vector space over R or C, the restriction of the Riemannian metric to \(f^{-1}(b)\times f^{-1}(b)\) gives a scalar product on the fibre \(f^{-1}(b))\). An endomorphism of the bundle \((E,p,B)\) is a pair of mappings \(X: E\to E\) and \(\chi: B\to B\), such that \(pX=\chi p\) and \(X|_{p^{-1}(b)}\) is a linear map of \(p^{-1}(b)\) to \(p^{- 1}(\chi b)\). Let \({\mathfrak M}=\{X_ t\}_{t\in M}\) be a family of endomorphisms parametrized by the set \(M\subset R.\) The set M is supposed to have \(+\infty\) as an accumulating point. The Lyapunov exponent is defined by \(\lambda ({\mathfrak M},\xi)=\limsup_{t\to +\infty,t\in M}| X_ t\xi | /t.\) Let \(\Theta\) (b) be the set of all bases of the fibre \(p^{-1}(b)\), whose elements \((\xi_ 1,...,\xi_ n)\) are numbered in order of non-increasing Lyapunov exponents \(\lambda({\mathfrak M},\xi_ 1)\geq...\geq \lambda({\mathfrak M},\xi_ n)\). A preordered relation on b is introduced, putting \((\xi_ 1,\xi_ 2,...,\xi_ n)>(\eta_ 1,\eta_ 2,...,\eta_ n)\) if and only if \(\lambda ({\mathfrak M},\xi_{{\dot \tau}})\geq \lambda ({\mathfrak M},\eta_{{\dot \tau}}),\) \({\dot \tau}=1,...,n\). A minimal bases with respect to this preordering is called normal. An existence and some extremal properties of the normal bases are established.