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

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 $\pi^{{\bf a}}:\; \Sigma={1,\ldots, m}^{\Bbb N}\to \Bbb R^d$ denote the coding map associated with the affine IFS ${T_ix+a_i}{i=1}^m$. We show that for every Borel probability measure $\mu$ on $\Sigma$, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of $\pi^{\bf a}\mu$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$, where $\pi^{\bf a}_\mu$ stands for the push-forward of $\mu$ by $\pi^{\bf a}$. In particular, we give a necessary and sufficient condition on $\mu$ so that $\pi^{\bf a}_*\mu$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Moreover, for every analytic set $E\subset \Sigma$, each of the Hausdorff, packing, lower and upper box-counting dimensions of $\pi^{{\bf a}}(E)$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.