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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   −  χ    }   , where   h   and   χ   are the entropy and Lyapunov exponent, respectively. The proof relies on Shmerkin's 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.

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