Dimension of ergodic measures projected onto self-similar sets with overlaps

For self-similar sets on R satisfying the exponential separation condition we show that the dimension of natural projections of shift invariant ergodic measures is equal to min{1,h-chi}, where h and chi are the entropy and Lyapunov exponent, respectively. The proof relies on Shmerkin's recent result on the Lq dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.