Hausdorff dimension of planar self-affine sets and measures

Abstract

Let $X=\bigcup {\phi }_{i}X$ be a strongly separated self-affine set in ${\mathbb{R}}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the ${\phi }_i$, 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.

47 pages, 2 figures