Local dimensions of overlapping self-similar measures

TL;DR

This paper investigates the local dimensions of certain self-similar measures, showing the existence of isolated points under specific overlapping conditions and providing bounds for Bernoulli convolutions.

Contribution

It demonstrates the presence of isolated local dimension points in overlapping self-similar measures and offers bounds for local dimensions in Bernoulli convolutions.

Findings

Existence of isolated local dimension points under overlap conditions.

Bounds on local dimensions for Bernoulli convolutions.

Application to convolution products of Bernoulli 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.