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.
Citation
Kathryn E. Hare. Kevin G. Hare. "Local Dimensions of Overlapping Self-Similar Measures." Real Anal. Exchange 44 (2) 247 - 266, 2019. https://doi.org/10.14321/realanalexch.44.2.0247
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