the so-called “transverse dimension” of these conditional measures, where significant efforts are made to handle the situation when the linear parts of the IFS do not satisfy any domination condition (in such case the angles of Oseledets subspaces maybe substantially small). Our strategy is to build an induced dynamics so that we are able to focus on the trajectories where the angles of Oseledets subspaces are significantly large.

Dimension formulas {#S-DF}
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Throughout this subsection, under the assumptions of Theorem \[thm-1.0\], we further assume that $m$ is ergodic. We are going to present certain dimension formulas for $\mu=m\circ \pi^{-1}$ and related conditional measures.

First notice that in this ergodic case, the condition (3) in Definition \[de-1.1\] is equivalent to $$\label{e-1.3}
\lambda:=\lim_{n\to \infty}\frac{1}{n}\int \log \|M_{x_0}\cdots M_{x_{n-1}}\|\; d m(x)<0.$$

By Oseledets’ multiplicative ergodic theorem [@Oseledets1968], there exist an integer $1\leq s\leq d$, numbers $\lambda=\lambda_1>\cdots>\lambda_s\geq -\infty$, positive integers $k_1, \ldots, k_s$ with $\sum_{i=1}^sk_i=d$, and measurable linear subspaces $$\R^d=V_x^0\supsetneq V_x^1\supsetneq\cdots\supsetneq  V_x^s=\{0\}, \quad x\in \Sigma,$$ such that for $m$-a.e. $x=(x_n)_{n=-\infty}^\infty$,

-   $M_{x_{-1}}V_x^i\subset V^i_{\sigma^{-1}x}$;

-   $\dim V_x^i=\sum_{j=i+1}^s k_j$;

-   $\lim_{n\to \infty} \frac{1}{n}\log \|M_{x_{-n}}\cdots M_{x_{-1}}v\|=\lambda_{i+1}$ for $v\in V_{x}^i\backslash V_{x}^{i+1}$.

The numbers $\lambda_1,\ldots, \lambda_s$ are called the [*Lyapunov exponents*]{} of $(M_j)_{j\in \Lambda}$ with respect to $m$, and $k_i$ 