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arXiv · 1807.08676

Local dimensions of overlapping self-similar measures

Kathryn E. Hare·Kevin G. Hare
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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.

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Kathryn E. Hare, Kevin G. Hare. 2018-07-23. Local dimensions of overlapping self-similar measures. https://arxiv.org/abs/1807.08676

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