Invariant measures for parabolic IFS with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the rea l line. An ergodic shift-invariant measure with positive entropy on the sym bolic space induces an invariant measure on the limit set of the IFS. The H ausdorff dimension of this measure equals the ratio of entropy over Lyapuno v exponent if the IFS has no "overlaps." We focus on the overlapping case a nd consider parameterized families of IFS, satisfying a transversality cond ition. Our main result is that the invariant measure is absolutely continuo us for a.e. parameter such that the entropy is greater than the Lyapunov ex ponent. If the entropy does not exceed the Lyapunov exponent, then their ra tio gives the Hausdorff dimension of the invariant measure for a.e. paramet er value, and moreover, the local dimension of the exceptional set of param eters can be estimated. These results are applied to a family of random con tinued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute con tinuity for a.e. parameter in some interval below the threshold.