The Furstenberg theorem: adding a parameter and removing the stationarity

The talk is based on a joint work with A. Gorodetski.
The classical Furstenberg Theorem describes the (almost sure) behaviour of a random product of independent matrices from $\mathrm{SL}(n,\mathbb{R})$; their norms turn out to grow exponentially. In our joint work [GK], we study what happens if the random matrices from $\mathrm{SL}(2,\mathbb{R})$ 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 generalisation for 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 Though in this setting most of the standard tools are no longer applicable (no more stationary measure, no more Birkhoff ergodic theorem, etc.), it turns out that the Furstenberg theorem can (under the appropriate assumptions) still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 generalised to this setting, with a deterministic sequence replacing the Lyapunov exponent. These two generalisations can be mixed together, providing the Anderson localisation conclusions for the non-stationary 1D random Schrödinger operators.
[GK] A. Gorodetski, V. Kleptsyn, Parametric Furstenberg Theorem on random products of $\mathrm{SL}(2,\mathbb{R})$ matrices, Advances in Mathematics, 378 (2021), 107522; preprint: arXiv:1809.00416.