Dimensions of orthogonal projections of typical self-affine sets and measures

Abstract

Let $T_1,\dots ,T_m$ be a family of $d\times d$ invertible real matrices with $\Vert T_i\Vert <1/2$ for $1\le i\le m$. For $\mathbf{a}=(a_1,\dots ,a_m)\in {\mathbb{R}}^{md}$, let ${\pi }^{\mathbf{a}}:\Sigma =\{1,\dots ,m{\}}^{\mathbb{N}}\to {\mathbb{R}}^d$ denote the coding map associated with the affine IFS $\{T_{i}x+a_i{\}}_{i=1}^m$, and let $K^{\mathbf{a}}$ denote the attractor of this IFS. Let $W$ be a linear subspace of ${\mathbb{R}}^d$ and $P_W$ the orthogonal projection onto $W$. We show that for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$, the Hausdorff and box-counting dimensions of $P_W(K^{\mathbf{a}})$ coincide and are determined by the zero point of a certain pressure function associated with $T_1,\dots ,T_m$ and $W$. Moreover, for every ergodic $\sigma$-invariant measure $\mu$ on $\Sigma$ and for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$, the local dimensions of $(P_W{\pi }^{\mathbf{a}}{)}_{\ast }\mu$ exist almost everywhere, here $(P_W{\pi }^{\mathbf{a}}{)}_{\ast }\mu$ stands for the push-forward of $\mu$ by $P_W{\pi }^{\mathbf{a}}$. However, as illustrated by examples, $(P_W{\pi }^{\mathbf{a}}{)}_{\ast }\mu$ may not be exact dimensional for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$. Nevertheless, when $\mu$ is a Bernoulli product measure, or more generally, a supermultiplicative ergodic $\sigma$-invariant measure, $(P_W{\pi }^{\mathbf{a}}{)}_{\ast }\mu$ is exact dimensional for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$.