Consider a finite sequence of linear contractions
${{S}_{j}}(x)\,=\,\rho x\,+{{d}_{j}}$
and probabilities
$pj\,>\,0$
with
$\Sigma \,{{p}_{j}}\,=\,1$
. We are interested in the self-similar measure
$\mu \,=\,\Sigma \,{{p}_{j}}\,\mu \,{}^\circ \,s_{j}^{-1}$
, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists a subset of supp
$\mu $
of full
$\mu $
and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.
To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the
$k$
-th convolution of the associated Cantor measure has local dimension at
$x\,\in \,(0,\,1)$
tending to 1 as
$k$
tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.