A family of measures associated with iterated function systems

Abstract

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$, $i=0,1,..., N-1$. 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)$ tends to zero as $ k\to \infty $, and if $p_{i}>0$ 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.