LAUSR

Let $$\mu $$μ be a measure on $${ SL}_{2}({\mathbb {R}})$$SL2(R) generating a non-compact and totally irreducible subgroup, and let $$\nu $$ν be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $$\mu $$μ is supported on finitely many matrices with algebraic entries, then $$\begin{aligned} \dim \nu =\min \Big \{1,\frac{h_{{{\mathrm{{\mathrm{RW}}}}}}(\mu )}{2\chi (\mu )}\Big \} \end{aligned}$$dimν=min{1,hRW(μ)2χ(μ)}where $$h_{{{\mathrm{{\mathrm{RW}}}}}}(\mu )$$hRW(μ) is the random walk entropy of $$\mu $$μ, $$\chi (\mu )$$χ(μ) is the Lyapunov exponent for the random matrix product associated with $$\mu $$μ, and $$\dim $$dim denotes pointwise dimension. In particular, for every $$\delta >0$$δ>0, there is a neighborhood U of the identity in $${ SL}_{2}(\mathbb {R})$$SL2(R) such that if a measure $$\mu \in \mathcal {P}(U)$$μ∈P(U) is supported on algebraic matrices with all atoms of size at least $$\delta $$δ, and generates a group which is non-compact and totally irreducible, then its stationary measure $$\nu $$ν satisfies $$\dim \nu =1$$dimν=1.