M}$, nowadays usually called the [*affinity dimension*]{} $\dim_{\rm AFF}({\bf M})$, which is always an upper bound for the upper box-counting dimension of $K({\bf M},\ba)$, and such that when $\|M_j\|<1/2$ for all $j$, then for ${\mathcal L}^{d|\Lambda|}$-a.e. $\ba$, $\dim_{\rm H}K({\bf M},\ba)=\min (d, \dim_{\rm AFF}({\bf M}))$. In fact, Falconer proved this with $1/3$ as the upper bound on the norms; it was subsequently shown by Solomyak [@Solomyak1998] that $1/2$ suffices.

The analogue of affinity dimension for measures is the Lyapunov dimension, which we denote $\dim_{\rm LY}(m, {\bf M})$; see Section \[S-7\] for its definition. In [@JordanPollicottSimon2007], Jordan, Pollicott and Simon proved that the Lyapunov dimension $\dim_{\rm LY}(m, {\bf M})$ is always an upper bound for the Hausdorff dimension of $m\circ (\pi_{\ba})^{-1}$, and moreover when $\|M_j\|<1/2$ for all $j$, then for ${\mathcal L}^{d|\Lambda|}$-a.e. $\ba$, $\dim_{\rm H} m\circ (\pi_{\ba})^{-1}=\min(d, \dim_{\rm LY}(m,{\bf M}))$.

Recall a set in a topological space is said to be of [*first category*]{} if it can be written as the countable union of nowhere dense subsets. As an application of Theorem \[thm-1.3\], we get the following result.

\[cor-1.7\] Suppose that $\|M_j\|<1/2$ for $j\in \Lambda$. Then the following hold.

-   For every ergodic $\sigma$-invariant measure $m$ on $\Sigma$, the exceptional set $$\left\{\ba\in 