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 $μ=\sum_{i}p_{i} μ_{i} \tag{*}$. 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.

14 pages including references. Corrections made on pp.4 and 13

References in corpus (1)

• Measures in wavelet decompositions