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