Dimensions of projected sets and measures on typical self-affine sets

Let $T_1,\dots ,T_m$ be a family of $d\times d$ invertible real matrices with $\mathrm{\parallel }{T}_i\mathrm{\parallel }<1\mathrm{/}2$ for $1\le i\le m$. For $\mathbf{a}=(a_1,\dots ,a_m)\in {\mathbb{R}}^{md}$, let ${\pi }^{\mathbf{a}}:\mathrm{\Sigma }=\{1,\dots ,m{\}}^{\mathbb{N}}\to {\mathbb{R}}^d$ denote the coding map associated with the affine IFS $\{T_{i}x+a_i{\}}_{i=1}^m$. We show that for every Borel probability measure $\mu$ on $\mathrm{\Sigma }$, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of ${\pi }_{\ast }^{\mathbf{a}}\mu$ is constant for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$, where ${\pi }_{\ast }^{\mathbf{a}}\mu$ stands for the push-forward of $\mu$ by ${\pi }^{\mathbf{a}}$. In particular, we give a necessary and sufficient condition on $\mu$ so that ${\pi }_{\ast }^{\mathbf{a}}\mu$ is exact dimensional for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$. Moreover, for every analytic set $E\subset \mathrm{\Sigma }$, each of the Hausdorff, packing, lower and upper box-counting dimensions of ${\pi }^{\mathbf{a}}(E)$ is constant for ${\mathcal{L}}^{md}$-a.e.~$\mathbf{a}\in {\mathbb{R}}^{md}$. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.