Local Dimensions of Overlapping Self-Similar Measures

Abstract

We show that any equicontractive, self-similar measure arising from the IFS of contractions \((S_{j})\), with self-similar set \([0,1]\), admits an isolated point in its set of local dimensions provided the images of \(S_{j}(0,1)\) (suitably) overlap and the minimal probability is associated with one (resp., both) of the endpoint contractions. Examples include \(m\)-fold convolution products of Bernoulli convolutions or Cantor measures with contraction factor exceeding \(1/(m+1)\) in the biased case and \(1/m\) in the unbiased case. We also obtain upper and lower bounds on the set of local dimensions for various Bernoulli convolutions.