Abstract

The paper is devoted to the problem of classification of linear cocycles up to cohomology. The main result is a theorem on the Jordan normal form saying that any linear cocycle is cohomologous to a block-triangular cocycle with irreducible block-conformal cocycles on the diagonal. Two invariants of cocycle cohomology, the algebraic hull and the set of invariant measures, and their interrelations are studied. We show that all random invariant measures of a cocycle are determined by the algebraic hull and, up to a cohomology, are deterministic. For orthogonal cocycles the two invariants are equivalent and they give a sub-relation of the equivalence relation of cocycle cohomology. A complete classification of the one- and two-dimensional linear cocycles is given. Our results are refinements of the multiplicative ergodic theorem of Oseledets, as we are able to describe the structure of a linear cocycle inside the invariant subspaces corresponding to different Lyapunov exponents.