  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.
