the [*multiplicity*]{} of $\lambda_i$, $i=1,\ldots,s$. Recall that $\pi(x)$ is well-defined for $m$-a.e. $x$. Hence there exists a Borel set $\Sigma'\subset\Sigma$ with $\sigma(\Sigma')=\Sigma'$ and $m(\Sigma')=1$ such that $\pi$ is well-defined on $\Sigma'$ and the above properties (i)-(iii) hold for $x\in \Sigma'$.

We remark that these linear subspaces $V_x^i$ only depend on $i$ and $x^{-}:=(x_j)_{j=-\infty}^{-1}$ since by (i)-(iii), one has $$V_{x}^i=\left\{v\in \R^d: \; \lim_{n\to \infty} \frac{1}{n}\log \|M_{x_{-n}}\cdots M_{x_{-1}}v\|\leq \lambda_{i+1}\right\}.$$ Using this property, we construct a family of measurable partitions $\xi_0,\ldots, \xi_s$ of $\Sigma'$ as follows: $$\xi_i(x):=\{y\in \Sigma':\; y^{-}=x^{-}, \; \pi y-\pi x\in V_x^i\},$$ here $\xi_i(x)$ is the $\xi_i$-atom that contains $x$ (see Sections \[Sect-con\] and \[S-4\] for the details). Moreover, let $$\label{e-pp}
\P=\{[j]_0\cap \Sigma':\; j\in \Lambda\}$$ be the canonical partition of $\Sigma'$, where $[j]_0:=\{x=(x_n)_{n=-\infty}^\infty\in \Sigma:\; x_0=j\}$. Define $$\label{e-hi1}
h_i=H_m(\P|\widehat{\xi_i}),\quad i=0,\ldots, s,$$ where $H_m(\cdot|\cdot)$ stands for the conditional entropy and $\widehat{\xi_i}$ is the $\sigma$-algebra generated by $\xi_i$ (see Section \[Sect-ent\] for the definitions).

Now we are ready to present the dimension formula for $m\circ \pi^{-1}$.

\[thm-1.1\] Let ${\mathcal S}=\{M_jx+a_j\}_{j\in \Lambda}$ be an affine IFS on $\R^d$ and $m$ be an ergodic $\sigma$-invariant measure on $\Sigma$. Suppose that ${\mathcal S}$ is average contracting with respect to $m$. Let $\mu=m\circ \pi^{-1}$. Then $$\label{ly-dim}
\dim_{\rm H}\mu=\sum_{i=0}^{s-1}\frac{h_{i+1}-h_i}{\lambda_{i+1}},$$ 