The investigation of the Lyapunov exponents of products of matrices are intensively investigated lately. This involves questions about the shape of the spectrum (in particular, the simplicity of the spectrum - is it typical for a matrix cocycle to have different Lyapunov exponents?) and about behaviour of certain measures. One should mention the works of Avila, Bochi, Bonatti, Viana.

In a work with Katrin Gelfert and Lorenzo Diaz we investigated a seemingly unrelated problem of Lyapunov spectrum of cocycles of circle diffeomorphisms (satisfying certain set of axioms). In this talk I'll explain, how this result can be applied to investigation of matrix cocycles.