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

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

$\dim\nu=\min\{1,\frac{h_{RW}(\mu)}{2\chi(\mu)}\}$

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.

- Last modified: 8/01/2018