  We study Smale skew product endomorphisms (introduced in [27]) now over
countable graph directed Markov systems, and we prove the exact dimensionality
of conditional measures in fibers, and then the global exact dimensionality of
the equilibrium measure itself. Our results apply to large classes of systems
and have many applications. They apply for instance to natural extensions of
graph-directed Markov systems. Another application is to skew products over
parabolic systems. We give also applications in ergodic number theory, for
example to the continued fraction expansion, and the backward fractions
expansion. In the end we obtain a general formula for the Hausdorff (and
pointwise) dimension of equilibrium measures with respect to the induced maps
of natural extensions $\mathcal T_\beta$ of $\beta$-maps $T_\beta$, for
arbitrary $\beta > 1$.
