Title: Sets of non-Lyapunov behaviour for Schroedinger cocycles

Abstract: We shall discuss the growth of transfer matrices of random Schroedinger operators in one dimension. While for a fixed value of the spectral parameter the rate of exponential growth is given by the Lyapunov exponent, this is almost surely not simultaneously true for all the values of the parameter. We shall survey the earlier results pertaining to the topological size of the exceptional set, and present some new ones pertaining to the metrical size. We shall also discuss the tight connection between the subject and the theory of Anderson localisation. Based on joint work with Ilya Goldsheid.