Hausdorff Dimension of Planar Self-Affine Sets and Measures with Overlaps

Abstract

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on ${\mathbb{RP}}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.

84 pages. v2: Correction to statement of Thm 4.1 (requires uniform entropy dimension, now defined in Def 3.14), and adjusted quantifier order in proof of Cor. 4.2. No changes to main results