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

## TL;DR

This paper establishes a formula for the dimension of ergodic measures projected onto self-similar sets with overlaps, using entropy and Lyapunov exponents, and extends the approach to convolutions and projections.

## Contribution

It provides a precise dimension formula for projections of ergodic measures on self-similar sets with overlaps, utilizing recent advances in $L^{q}$ dimensions.

## Key findings

- Dimension of projected measures equals $rac{h}{-h}$ under certain conditions
- Results on convolutions of ergodic measures
- Results on orthogonal projections of ergodic measures

## 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.