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

Abstract

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