On the dimension of Furstenberg measure for $SL_2(R)$ random matrix products

Abstract

Let $\mu$ be a measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $\chi>0$ denote its Lyapunov exponent, and let $\nu$ be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $\mu$ is supported on finitely many matrices with algebraic entries, then \[ \dim\nu=\min\{1,\frac{h_{\textrm{RW}}(\mu)}{2\chi}\} \] where $h_{\textrm{RW}}(\mu)$ is the random walk entropy of $\mu$, and $\dim$ denotes pointwise dimension. In particular, for every $\delta>0$, there is a neighborhood $U$ of the identity in $SL_{2}(\mathbb{R})$ such that if a measure $\mu\in\mathcal{P}(U)$ is supported on algebraic matrices 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 $\dim\nu=1$.