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

For self-similar sets on $mathbb{R}$ satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to $min{1,frac{h}{-chi}}$, where $h$ and $chi$ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkins recent result on the $L^{q}$ 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.