Doubling properties of self-similar measures

Let {F_i}_iⁿ be a system of similitudes in ℝⁿ. We study necessary and sufficient conditions for their associated selfsimilar measures to be doubling on its support. An equivalent condition is obtained when {F_i} satisfies the open set condition. The condition allows us to construct many examples of interest. In the case where the open set condition is not satisfied, we study an infinitely convoluted Bernoulli measure (associated with the golden ratio ρ = ( √5 -1 ) / 2) and give a necessary and sufficient condition for it to be doubling on its support [0,1], Indiana University Mathematics Journal