On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average

Suppose {f 1,...,f m } is a set of Lipschitz maps of 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 ((sigma) ^m i=1 p i =1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on d and as a corollary show that the measure will be singular if the modulus of the entropy (sigma)i p i log p i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannonʹʹs Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of R.