for $\delta^u$ and $\delta^s$ in terms of conditional entropies and Lyapunov exponents, which nowadays is called “Ledrappier-Young formula". Eckmann and Ruelle [@EckmannRuelle1985] indicated that it is unknown whether the local dimension of $\mu$ is equal to the sum of $\delta^u$ and $\delta^s$ if $\mu$ is a hyperbolic measure. Then the problem was referred as Eckmann-Ruelle conjecture, and was finally answered confirmatively by Barreira, Pesin and Schmeling in 1999 for $C^{1+\alpha}$ diffeomorphisms [@BarreiraPesinSchmeling1999]. Later, the result of exact dimensionality was further extended by Qian and Xie [@QianXie2008] and Shu [@Shu2010] to $C^2$ expanding endomorphisms and $C^2$ non-degenerate endomorphisms, respectively.

For the study of IFSs, it is well-known that if ${\mathcal S}$ is a contractive IFS consisting of similarity maps, or more generally, a contracting $C^1$ conformal IFS, then under an additional separation condition (the so-called open set condition), the push-forward measure of any ergodic invariant measure under the coding map is exact dimensional with dimension given by the classical entropy divided by the Lyapunov exponent (cf. [@Bedford1991; @Hutchinson1981; @Patzschke1997]). The result essentially follows from the Shannon-McMillan-Breiman theorem in entropy theory. However, the problem becomes much more complicated without assuming the open set condition. In [@FengHu2009], by introducing a notion of projection 