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

Abstract

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 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.

16 pages. A few details have been clarified and section 5 has been expanded. Now accepted in the Proceedings of the LMS