On the dimension of Furstenberg measure for $SL_2(\mathbb{R})$ random matrix products

Mon, 15/01/2018 - 14:00

Let $\mu$ be a finitely-supported measure on $SL_{2}(\mathbb{R})$ generating a non-compact and 
totally irreducible subgroup, and let $\nu$ be the associated stationary (Furstenberg) measure. 
We prove that if the support of $\mu$ is ``Diophantine,'' then
where $h_{RW}(\mu)$ is the random walk entropy of $\mu$, $\dim$ denotes pointwise dimension, 
and $\chi$ is the Lyapunov exponent of the random walk generated by $\mu$.
In particular, for every $\delta>0$, there is a neighborhood $U$ of the identity in 
$SL_{2}(\mathbb{R})$ such that if $\mu$ has support in $U$ on matrices with algebraic entries, 
is atomic with all atoms of size at least $\delta$, and generates a group which is non-compact 
and totally irreducible, then its stationary measure $\nu$ satisfies  $\dim\nu=1$.
This is a joint work with M. Hochman.
In my talk, I will try to explain the concepts and motivate the result.