The Furstenberg theorem : adding a parameter and removing the stationarity (On a joint work with A. Gorodetski)

The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices from SL(n,R); their norms turn out to grow exponentially. In our joint work, we study what happens if the random matrices from SL(2,R) depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some natural conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.
Another direction of the generalization of the classical Furstenberg theorem is removing the stationarity assumption. That is, the matrices that are multiplied are still independent, but no longer identically distributed. Though in this setting most of the standard tools are no longer applicable (no more stationary measure, no more Birkhoff ergodic theorem, etc.), it turns out that the Furstenberg theorem can (under the appropriate assumptions) still be generalized to this setting, with a deterministic sequence replacing the Lyapunov exponent. These two generalizations can be mixed together, providing the Anderson localization conclusions for the non-stationary 1D random Schrodinger operators.