Hausdorff dimension of planar self-affine sets and measures

TL;DR

This paper proves that the Hausdorff dimension of certain planar self-affine sets and measures equals their affinity and Lyapunov dimensions under mild conditions, using projection analysis and additive combinatorics.

Contribution

It establishes the equality of Hausdorff and affinity dimensions for a broad class of planar self-affine sets and measures under mild assumptions.

Findings

Hausdorff dimension equals affinity dimension for self-affine sets.

Dimension of self-affine measures matches Lyapunov dimension.

Uses projection analysis and additive combinatorics techniques.

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.