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

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

Information

Published: 2019
First available in Project Euclid: 1 May 2020

zbMATH: 07211591
Digital Object Identifier: 10.14321/realanalexch.44.2.0247

Subjects:
Primary: 28C15
Secondary: 28A80 , 37C45

Keywords: Bernoulli convolution , Cantor measure , local dimension

Rights: Copyright © 2019 Michigan State University Press

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Vol.44 • No. 2 • 2019
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