Sets of non-Lyapunov behaviour for matrix cocycles

A matrix cocycle is a non-commutative counterpart of random walk. The counterpart of the ergodic theorem, describing the almost sure asymptotic behaviour to leading order, is given by the theory of random matrix products originating in the works of Furstenberg—Kesten, Furstenberg, and Oseledec. On the other hand, the spectral theory of random one-dimensional second-order operators leads to the study of cocycles depending on an additional real number (the spectral parameter), and, a priori, the theory is applicable for almost all (rather than all) values of the parameter. The focus of the talk will be on the exceptional sets, where different asymptotic behaviour occurs: particularly, we shall discuss their rôle in spectral theory and their topologic and metric properties, including a result resembling the Jarnik theorem on Diophantine approximation. Based on joint work with Ilya Goldsheid.