Consider a random product of i.i.d. matrices, randomly chosen from SL(2,R), satisfying some reasonable nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric).Then a classical Furstenberg theorem implies that the norm of such a random product almost surely grows exponentially.
Now, what happens if each of these matrices depends on an additional parameter? We will discuss the case when the dependence of angle is monotonous w.r.t. the parameter: increasing the parameter «rotates all the directions clockwise».
It turns out that (under some reasonable conditions)
- Almost surely for all the parameter values, except for a zero Hausdorff dimension (random) set, the Lyapunov exponent exists and equals to the Furstenberg one.
- Almost surely for all the parameter values the upper Lyapunov exponent equals to the Furstenberg one
- At the same time, in the no-uniform-hyperbolicity parameter region there exists a dense subset of parameters, for each of which the lower Lyapunov exponent takes any fixed value between 0 and the Furstenberg exponent.
The talk is based on a joint project with A. Gorodetski.