\R^{d|\Lambda|}:\; \dim_{\rm H} \left(m\circ (\pi_{\ba})^{-1}\right)\neq \min(d, \dim_{\rm LY}(m, {\bf M}))\right\}$$ is of first category in $\R^{d|\Lambda|}$.

-   The exceptional set $$\left\{\ba\in \R^{d|\Lambda|}:\; \dim_{\rm H} K({\bf M}, \ba)\neq \min(d, \dim_{\rm AFF}({\bf M}))\right\}$$ is of first category in $\R^{d|\Lambda|}$.

The above result says that these exceptional sets are also small in a topological sense. A fundamental and challenging question is to specify those translation vectors not lying in the exception sets. Significant progresses have been made recently in [@Barany2015; @FalconerKempton2018; @BaranyKaenmaki2015; @Rapaport2015], showing that under certain additional assumptions, the Hausdorff and Lyapunov dimensions of a self-affine measure (or more generally, the push-forward of a quasi-Bernoulli measure) coincide if the involved Furstenberg measures have enough large dimension. In next theorem we will drop off some redundant assumptions used in these works and further extend the result to the push-forward measures of ergodic sub-multiplicative measures.

Recall that for a Borel probability measure $\eta$ on a metric space, its [*upper Hausdorff dimension*]{} $\dim_{\rm H}^*\eta$ is the smallest Hausdorff dimension of a Borel set $F$ of $\eta$ measure $1$. Set $d_0=0$ and $d_\ell=k_1+\cdots+k_\ell$ for $1\leq \ell\leq s$.

\[thm-1.4\] Let $\mathcal S=\{M_jx+a_j\}_{j\in \Lambda}$ be a contracting affine IFS on $\R^d$ satisfying the strong separation condition and $m\in 