the plane. In [@KenyonPeres1996], Kenyon and Peres extended Bedford and McMullen’s result to higher dimensional self-affine carpets, and moreover, they proved the exact dimensionality and gave a Ledrappier-Young type dimension formula for arbitrary ergodic invariant measure on these carpets. For more related results on carpet-like self-affine sets, see the survey paper [@Falconer2013].

In [@FengHu2009], Feng and Hu proved that for each contracting invertible affine IFS in $\R^d$, Theorem \[thm-1.0\] holds under an additional assumption that the linear parts of the IFS commute (i.e. $M_iM_j=M_jM_i$). It remained open whether this additional assumption could be removed. Very recently, Bárány and Käenmäki [@BaranyKaenmaki2015] made a substantial progress. They proved that for contracting invertible affine IFSs, every planar self-affine measure (more generally, every self-affine measure in $\R^d$ having $d$ distinct Lyapunov exponents) is exact dimensional, and moreover, under certain domination condition on the linear parts $\{M_j\}$, the push-down of every quasi-Bernoulli measure on the self-affine set is exact dimensional, with dimension given by a Ledrappier-Young type formula. Some other partial results were also obtained in [@Barany2015; @Rapaport2015; @FraserJordanJurga2017]. Along another direction, it is proved that for a given ergodic $m\in {\mathcal M}_{\sigma}(\Sigma)$, $m\circ \pi^{-1}$ is exact dimensional for “almost all” contracting invertible affine IFSs 