Hausdorff dimension of planar self-affine sets and measures

Bárány, Balázs and Hochman, Michael and Rapaport, Ariel (2019) Hausdorff dimension of planar self-affine sets and measures. INVENTIONES MATHEMATICAE, 216 (3). pp. 601-659. ISSN 0020-9910 (print), 1432-1297 (online)

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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.

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