On the fine structure of stationary measures in systems which contract-on-average

Abstract

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 $\mathbb{R}$.

Final version; 14 pages in Latex