IFS on $\R^d$ and $m\in \M_\sigma(\Sigma)$. Suppose that ${\mathcal S}$ is average contracting with respect to $m$. Let $\mu=m\circ \pi^{-1}$. Then

-   $\dim_{\rm loc}(\mu, x)$ exists for $\mu$-a.e. $x\in \R^d$.

-   Assume furthermore that $m$ is ergodic. Then $\mu$ is exact dimensional and $\dim_{\rm H}\mu$ satisfies a Ledrappier-Young type dimension formula.

The precise dimension formula of $\mu$ and some of its applications will be given in Sections \[S-DF\]-\[S-SC\]. Below we first give some background information about the above study.

The problem of the existence of local dimensions has a long history in smooth dynamical systems, as well as in the study of IFSs. It is of great importance in dimension theory of dynamical systems and fractal geometry. In [@Young1982], Young proved that an ergodic hyperbolic measure invariant under a $C^{2}$ surface diffeomorphism is always exact dimensional. (Here by hyperbolic one means that the measure has no zero Lyapunov exponent.) For a hyperbolic measure $\mu$ in high-dimensional $C^{2}$ systems, Ledrappier and Young [@LedrappierYoung1985] proved the existence of $\delta^u$ and $\delta^s$, the local dimensions along stable and unstable local manifolds, respectively, and the upper local dimension of $\mu$ is bounded by the sum of $\delta^u$ and $\delta^s$; moreover they obtained a formula 