Dimensions of orthogonal projections of typical self-affine sets and measures
De-Jun Feng, Yu-Hao Xie
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 $π^{\bf a}\colon Σ=\{1,\ldots, m\}^{\Bbb N}\to {\Bbb R}^d$ 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 $σ$-invariant measure $μ$ on $Σ$ and for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the local dimensions of $(P_Wπ^{\bf a})_*μ$ exist almost everywhere, here $(P_Wπ^{\bf a})_*μ$ stands for the push-forward of $μ$ by $P_Wπ^{\bf a}$. However, as illustrated by examples, $(P_Wπ^{\bf a})_*μ$ may not be exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$. Nevertheless, when $μ$ is a Bernoulli product measure, or more generally, a supermultiplicative ergodic $σ$-invariant measure, $(P_Wπ^{\bf a})_*μ$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$.