On the dimension of Furstenberg measure for SL₂(R) random matrix products

Let μ be a measure on SL₂(R) generating a non-compact and totally irreducible subgroup, and let ν be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if μ is supported on finitely many matrices with algebraic entries, then (Formula Presented.)where h_RW(μ) is the random walk entropy of μ, is the Lyapunov exponent for the random matrix product associated with μ, and dim denotes pointwise dimension. In particular, for every δ> 0 , there is a neighborhood U of the identity in SL₂(R) such that if a measure μ∈ P(U) is supported on algebraic matrices with all atoms of size at least δ, and generates a group which is non-compact and totally irreducible, then its stationary measure ν satisfies dim ν= 1.