\mathcal M_\sigma(\Sigma)$ be ergodic and sub-multiplicative. Let $i$ be the unique element in $\{1,\ldots, s\}$ so that $d_{i-1}  \leq \dim_{\rm LY} (m, {\bf M})<d_i$. Then $$\label{e-tz}
\dim_{\rm H} m\circ \pi^{-1}=\dim_{\rm LY}(m, {\bf M})$$ provided one of the following conditions holds:

-   $s=1$.

-   $i=s>1$, $\lambda_s\neq -\infty$ and $$\label{e-condLY}
     \dim_{\rm H}^*\left(m\circ (\Pi_{s-1})^{-1}\right) + \dim_{\rm LY} (m, {\bf M})\geq  d_{s-1} (d-d_{s-1}+1).$$

-   $1\leq i \leq s-1$, and $$\begin{aligned}
            &\dim_{\rm H}^*\left(m\circ (\Pi_{i})^{-1}\right)-\dim_{\rm LY}(m, {\bf M})\geq d_{i}(d-d_i-1), \label{e-al2}\\
        &\dim_{\rm H}^*\left(m\circ (\Pi_{i-1})^{-1}\right) + \dim_{\rm H} m\circ \pi^{-1} \geq d_{i-1}(d-d_{i-1}+1). \label{e-al1}
        \end{aligned}$$

The conditions (b), (c) in the above theorem were introduced in [@Rapaport2015] and [@BaranyHochmanRapaport2017], respectively, in slightly stronger forms. For a contracting invertible affine IFS, Rapaport [@Rapaport2015] proved the implication (b)$\Rightarrow$ in the case when $m$ is Bernoulli and $(M_j)_{j\in \Lambda}$ satisfies an irreducibility assumption; whilst Bárány and Käenmäki [@BaranyHochmanRapaport2017] proved under the assumptions that the conditions - hold for all $i\in \{1,\ldots, s-1\}$, $m$ is Bernoulli and $d=2$, or $m$ is quasi-Bernoulli and $\{M_j\}_{j\in \Lambda}$ satisfies a domination condition.

We remark that always holds whenever $i=1$, since $d_0=0$. It is worth pointing out that for every affine IFS $\mathcal S=\{M_jx+a_j\}_{j\in \Lambda}$ on $\R^d$, there exists at least one ergodic $m\in \mathcal M_\sigma(\Sigma)$, called 