  We study skew product lifts and overlap numbers for equilibrium measures
\mu_\psi of H\"older continuous potentials \psi on such lifts. We find
computable formulas and estimates for the overlap numbers in several concrete
significant cases of systems with overlaps. In particular we obtain iterated
systems which are asymptotically irrational-to-1 and absolutely continuous on
their limit sets. Then we look into the general structure of the Rokhlin
conditional measures of \mu_\psi with respect to different fiber partitions
associated to the lift \Phi, and find relations between them. Moreover we prove
an estimate on the box dimension of a certain associated invariant measure
\nu_\psi on the limit set \Lambda by using the overlap number of \mu_\psi.
