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

Let mu be a self-affine measure on R-d associated to an affine IFS Phi and a positive probability vector p. Suppose that the maps in Phi do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that dim mu is equal to the Lyapunov dimension dim(L)(Phi, p) whenever d = 3 and Phi satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring dim mu = min{d, dim(L)(Phi, p)}, from which earlier results in the planar case also follow. Additionally, we prove that dim mu = d whenever Phi is Diophantine (which holds e.g. when Phi is defined by algebraic parameters) and the entropy of the random walk generated by Phi and p is at least (chi(1) - chi(d))(d-1)(d-2)/2 - Sigma(d)(k=1) chi(k), where 0 > chi(1) >= ... >= chi(d) are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of mu. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.