The dimension of projections of self-affine sets and measures

Abstract

Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if μ is a Bernoulli measure on E with dimH μ = dimL μ, where dimH and dimL denote Hausdorff and Lyapunov dimensions, then the projection of μ in all but at most one direction has Hausdorff dimension min(dimH μ, 1). We transfer this result to sets and show that many self-affine sets have projections of dimension min(dimH E, 1) in all but at most one direction.