  Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We
form the iterated function system (IFS) by independently choosing the maps so
that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We
assume that the IFS contracts on average. We give an upper bound for the
Hausdorff dimension of the invariant measure induced on $\mathbb{R}^d$ and as a
corollary show that the measure will be singular if the modulus of the entropy
$\sum_i p_i \log p_i$ is less than $d$ times the modulus of the Lyapunov
exponent of the system. Using a version of Shannon's Theorem for random walks
on semigroups we improve this estimate and show that it is actually attainable
for certain cases of affine mappings of $\mathbb{R}$.
