The pointwise dimension of self-similar measures in complete metric spaces (Q1772603)

For a probability measure \(\mu\) on a complete metric space \(X\), the pointwise dimension of \(\mu\) at \(x \in X\) is given by \[ d_\mu(x):=\lim_{r \to 0} \frac{\log \mu(B(x,r))}{\log r} \] provided that the limit exists. In the paper, the pointwise dimension of self-similar measures is investigated under the condition that the strong open set condition is satisfied.