  We consider one-parameter families of smooth uniformly contractive iterated
function systems $\{f^\lambda_j\}$ on the real line. Given a family of
parameter dependent measures $\{\mu_{\lambda}\}$ on the symbolic space, we
study geometric and dimensional properties of their images under the natural
projection maps $\Pi^\lambda$. The main novelty of our work is that the
measures $\mu_\lambda$ depend on the parameter, whereas up till now it has been
usually assumed that the measure on the symbolic space is fixed and the
parameter dependence comes only from the natural projection. This is especially
the case in the question of absolute continuity of the projected measure
$(\Pi^\lambda)_*\mu_\lambda$, where we had to develop a new approach in place
of earlier attempt which contains an error. Our main result states that if
$\mu_\lambda$ are Gibbs measures for a family of H\"older continuous potentials
$\phi^\lambda$, with H\"older continuous dependence on $\lambda$ and
$\{\Pi^\lambda\}$ satisfy the transversality condition, then the projected
measure $(\Pi^\lambda)_*\mu_\lambda$ is absolutely continuous for Lebesgue
a.e.\ $\lambda$, such that the ratio of entropy over the Lyapunov exponent is
strictly greater than $1$. We deduce it from a more general almost sure lower
bound on the Sobolev dimension for families of measures with regular enough
dependence on the parameter. Under less restrictive assumptions, we also obtain
an almost sure formula for the Hausdorff dimension. As applications of our
results, we study stationary measures for iterated function systems with
place-dependent probabilities (place-dependent Bernoulli convolutions and the
Blackwell measure for binary channel) and equilibrium measures for hyperbolic
IFS with overlaps (in particular: natural measures for non-homogeneous
self-similar IFS and certain systems corresponding to random continued
fractions).
