Abstract: In this talk we will focus on an extended family of invariant measures of Iterated Function Systems (IFS). Namely, the corresponding probabilities are Hölder continuous maps. We are going to show that if the IFS satisfies the so-called transversality condition for a set of parameters then the Hausdorff dimension of the measure is the ratio of the entropy and the Lyapunov exponent for Lebesgue almost every parameters. Moreover, whenever the ratio is strictly greater than 1 the measure is absolutely continuous for Leb.-a.e. parameters. We will demonstrate our results and proofs on Bernoulli convolutions.