A family of measures associated with iterated function systems

Let $(X,d)$ be a compact metric space, and let an iterated function system (IFS) be given on $X$, i.e., a finite set of continuous maps ${\sigma }_i$: $X\to X$, $i=0,1,\mathrm{.}\mathrm{.}\mathrm{.},N-1$. The maps ${\sigma }_i$ transform the measures $\mu$ on $X$ into new measures ${\mu }_i$. If the diameter of $ \sigma_{i_{1}}\circ

… \circ \sigma_{i_{k}}(X)$tendstozeroas$ k\to \infty $,andif$p_{i}>0$satisfies$\sum_{i}p_{i}=1$, then it is known that there is a unique Borel probability measure $\mu$ on $X$ such that $\mu =\sum_{i}p_{i} \mu_{i} \tag{}$. In this paper, we consider the case when the $p_i$s are replaced with a certain system of sequilinear functionals. This allows us to study the variable coefficient case of (), and moreover to understand the analog of (*) which is needed in the theory of wavelets.