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.

中文翻译：

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遍历测度的维数投影到具有重叠的自相似集合上

对于满足指数分离条件的$ \ mathbb {R} $上的自相似集，我们证明了位移不变遍历测度的自然投影等于$ \ min \ {1，\ frac {h} {-\ chi} \ } $，其中$ h $和$ \ chi $分别是熵和Lyapunov指数。证明依赖于Shmerkin最近在自相似度量的$ L ^ {q} $维度上得出的结果。我们还使用相同的方法来给出遍历和正交投影到遍历测度的结果，这些遍历测度被投影到自相似集合上。