Typical property of Lyapunov exponents (Q1088832)

In this paper locally trivial vector bundles with fiber \(R^ n\) are considered. A Riemannian metric is introduced, and properties of parametrized families of endomorphisms are studied (a special case is a homomorphism of the additive semigroup \(R^+\) (or \(Z^+)\) into the multiplicative semigroup of endomorphisms of the vector bundle). The Lyapunov exponents of a family of endomorphisms at a point of the space of the vector bundle are defined. The main results are that the Lyapunov exponents are functions of the second Baire class defined on the base of the vector bundle, and that on the base all Lyapunov exponents of a family are generically upper semicontinuous.