**Paulo Varandas**

*Departamento de Matemática, Universidade Federal da Bahia, Salvador, Brazil & CMUP, University of Porto - Portugal*

The foundations of ergodic theory rely on ergodic theorems due to von Neumann, Birkhoff, Kingman, Hopf among others, which are extensions of the law of large numbers to the dynamical systems framework. In rough terms, these ensure that the time averages of a real valued function converge almost everywhere with respect to any invariant measure, and that its average coincides with the space average of the observable function.

Among some more sophisticated ergodic theorems, for functions taking values on non-abelian groups, one can find the celebrated Oseledets' theorem, which allow to define Lyapunov exponents (the natural extension of the concept of eigenvalues) for random products of matrices.

In the context of random products of matrices, we investigate both the set of points for which Lyapunov exponents are not well defined, and the set of directions along which such erratic behavior can be attained. The analysis will reduce to the study of iterated function systems which fail to satisfy any of the specification or gluing orbit properties.

In this talk I will mainly focus on the motivation and geometric mechanisms used to generate Lyapunov ``non-typical" behavior.

This is a joint work with G. Ferreira (UFMA-Brazil).