[*Käenmäki measure*]{}, so that $\dim_{\rm LY}(m, {\bf M})=\dim_{\rm AFF}({\bf M})$. This was first proved by Käenmäki [@Kaenmaki2004] in the case when $\mathcal S$ is invertible, and it extends to the general case by the sub-additive thermodynamic formalism [@CaoFengHuang2008]. Very recently, Bochi and Morris [@BochiMorris2018] showed that whenever $\mathcal S$ is invertible, each Käenmäki measure is sub-multiplicative. Hence for an invertible ${\mathcal S}$ satisfying the strong separation condition, if one of the conditions (a)-(c) in Theorem \[thm-1.4\] fulfills for some Käenmäki measure $m$, then $\dim_{\rm H} K({\bf M}, {\bf a})=\dim_{\rm AFF}({\bf M})=\dim_{\rm  H}m\circ \pi^{-1}$.

To check the conditions (b)-(c) in Theorem \[thm-1.4\], one needs to estimate the (upper) Hausdorff dimension of Furstenberg measures $m\circ (\Pi_{i})^{-1}$. So far there have been only a few dimensional results on these measures. In the case $d=2$, Hochman and Solomyak [@HochmanSolomyak2017] calculated the Hausdorff dimension of Furstenberg measures for Bernoulli $m$ under some mild assumptions. In [@BRS2018 Sect. 2.4], Bárány, Rams and Simon determined the Hausdorff dimension of Furstenberg measures for some special triangular affine IFSs in $\R^d$, in which $m$ could be any ergodic measure.

We remark that the conditions of Theorem \[thm-1.4\] might not be sharp. Very recently, Bárány, Hochman and Rapaport [@BaranyHochmanRapaport2017] made 