average contracting with respect to an ergodic $m\in {\mathcal M}_\sigma(\Sigma)$. Let $\mu=m\circ \pi^{-1}$. Assume $s\geq 2$ and in addition that $M_jM_{j'}=M_{j'}M_j$ for $j,j'\in \Lambda$. Then for $i\in \{1,\ldots, s-1\}$, $V_x^i$ is constant $m$-a.e., denoted by $W_i$, moreover, $\mu$ is dimension conserving with respect to $P_{(W_i)^\perp}$.

It is worth pointing out that if $\mu$ is a contracting self-similar measure in $\R^d$ with a finite rotation group, then for each proper subspace $W$ of $\R^d$, $\mu$ is dimension conserving with respect to $P_W$. The result is due to Falconer and Jin [@FalconerJin2014]. Under an additional assumption of the strong separation condition, this result can be alternatively derived from a general result of Furstenberg (cf. [@Furstenberg2008 Theorem 3.1]). We remark that this dimension conservation property also extends to ergodic invariant measures for rotation-free self-similar IFSs (see Remark \[rem-6.3\]). Furthermore, we remark that Theorem \[cor-1.0\]-\[thm-1.7’\] can be applied to analyze slices and projections of certain self-affine sets (see Remark \[rem-6.4\]).

Semi-continuity of dimension and applications {#S-SC}
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Here we present a semi-continuity result on the dimension of ergodic invariant measures for affine IFSs and give its application to the dimension of self-affine sets. Again let $\mathcal S=\{M_ix+a_i\}_{i\in \Lambda}$ be an affine IFS on $\R^d$, 