Monday, October 15, 2018 at 4:15pm to 5:15pm
Victor Kleptsyn (Institute of Mathematical Research of Rennes)
Furstenberg theorem: now with a parameter!
My talk will be devoted to our joint work with Anton Gorodetski.
Consider a random product of i.i.d. matrices, randomly chosen from SL(2,R): T_n=A_n ... A_2 A_1, where the random matrices A_i are i.i.d.
A classical Furstenberg theorem then implies, that under some very mild nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric) for the law of $A_i$'s the norm of such a product almost surely grows exponentially.
Now, what happens if each of these matrices $A_i(s)$ depends on an additional parameter $s$, and hence so does their product $T_n(s)$? For each individual $s$, the Furstenberg theorem is still applicable. However, what can be said almost surely for the random products $T_n(s)$, depending on a parameter?
We will impose a few reasonable additional assumptions, of which the most important is that the dependence of angle is monotonous w.r.t. the parameter: increasing the parameter «rotates all the directions clockwise».
It turns out that, under these assumptions,
- 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.
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof.