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

Let T1,..., Tmbe a family of d xdinvertible real matrices with Ti < 1/2for 1 = i = m. For a=( a1,..., am). Rmd, let pa: S ={1,..., m}N. Rddenote the coding map associated with the affine IFS {Tix + ai} mi= 1. We show that for every Borel probability measure mu on S, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of pa*mu is constant for Lmda.e. a. Rmd, where pa* mu stands for the push-forward of mu by pa. In particular, we give a necessary and sufficient condition on mu so that pa*mu is exact dimensional for Lmd-a.e. a. Rmd. Moreover, for every analytic set E. S, each of the Hausdorff, packing, lower and upper box-counting dimensions of pa(E) is constant for Lmd-a.e. a. Rmd. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated. (c) 2023 Elsevier Inc. All rights reserved.