On self-affine measures associated to strongly irreducible and proximal systems

Let μ be a self-affine measure on Rd associated to an affine IFS Φ and a positive probability vector p. Suppose that the maps in Φ do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that dim⁡μ is equal to the Lyapunov dimension dimL⁡(Φ,p) whenever d=3 and Φ satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring dim⁡μ=min⁡{d,dimL⁡(Φ,p)}, from which earlier results in the planar case also follow. Additionally, we prove that dim⁡μ=d whenever Φ is Diophantine (which holds e.g. when Φ is defined by algebraic parameters) and the entropy of the random walk generated by Φ and p is at least (χ1−χd)(d−1)(d−2)2−∑k=1dχk, where 0>χ1≥...≥χd are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of μ.