average contracting with respect to an ergodic invariant measure $m$ on $\Sigma$. Write $\ba=(a_i)_{i\in \Lambda}$. To emphasize the dependence on $\ba$, let $\pi_\ba$ be the coding map associated to $\mathcal S$ and let $h_{i,\ba}$ ($i=1,\ldots, s$) be the conditional entropies of $m$ defined in . Then we have the following.

\[thm-1.3\]

-   The mapping $\ba\mapsto h_{i,\ba}$ is upper semi-continuous for each $i\in \{1,\ldots, s\}$.

-   Moreover, the mapping $\ba\mapsto \dim_{\rm H} \left(m\circ (\pi_{\ba})^{-1}\right)$ is lower semi-continuous.

Part (1) of the above result was first proved by Rapaport [@Rapaport2015 Lemma 8] in the case when $m$ is a Bernoulli product measure and ${\mathcal S}$ is invertible and contracting. Part (2) was shown by Hochman and Shmerkin [@HochmanShmerkin2012] for a special class of self-similar measures on $\R$. In Remark \[rem-8.1\] we give a further extension of Theorem \[thm-1.3\].

Below we present an application of Theorem \[thm-1.3\] to the dimension of self-affine sets and associated stationary measures. For this purpose, in the remaining part of this subsection we assume that $\|M_j\|<1$ for $j\in \Lambda$ and write ${\bf M}=(M_j)_{j\in \Lambda}$. Let $K({\bf M},\ba)$ be the self-affine set generated by the IFS ${\mathcal S}=\{M_jx+a_j\}_{j\in \Lambda}$. In 1988, Falconer [@Falconer1988] introduce a quantity associated to ${\bf 