Hausdorff dimension of planar self-affine sets and measures

By Balázs Bárány, Michael Hochman, Ariel Rapaport. Inventiones Mathematicae, 2017

126 citations recorded by the source corpus.

Let $$X={\bigcup }{\varphi }_{i}X$$X=⋃φiX be a strongly separated self-affine set in $${\mathbb {R}}^2$$R2 (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the $$\varphi _{i}$$φi, we prove that $$\dim X$$dimX 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/db4246d3bf14e02c1aadf585b374cce64783d51e

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Citations in the corpus

Conformality for a robust class of non-conformal attractors (2019, background): “Let us also note that only very recently Bárány-Hochman-Rapaport [2] provided a complete answer for Iterated-Function-Systems on the plane.”
Dimension of invariant measures for affine iterated function systems (2019, background): “For a contracting invertible affine IFS, Rapaport [59] proved the implication (b)⇒ (1.11) in the case when m is Bernoulli and (Mj)j∈Λ satisfies an irreducibility assumption; whilst Bárány and Käenmäki [3] proved (1.11) under the assumptions that the conditions (1.13)-(1.14) hold for all i ∈ {1, ”