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arXiv · 1712.07353

Hausdorff dimension of planar self-affine sets and measures

Balázs Bárány·Michael Hochman·Ariel Rapaport
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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-compactness 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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Balázs Bárány, Michael Hochman, Ariel Rapaport. 2017-12-20. Hausdorff dimension of planar self-affine sets and measures. https://arxiv.org/abs/1712.07353

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