Hausdorff dimension of planar self-affine sets and measures

Abstract

Let $$X={\bigcup }{\varphi }_{i}X$$ be a strongly separated self-affine set in $${\mathbb {R}}^2$$ (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the $$\varphi _{i}$$ , we prove that $$\dim X$$ is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.