ABSTRACT: I will study the invariant measures of systems of iterated functions (IFS) composed of affine maps. Unlike the scholastic examples, where a finite number of maps is considered and unlike some countable extensions thereof,  these maps belong to an uncountable set. I will describe the simplest case where such uncountable set is parameterized by a few real parameters, with a continuous (and possibly, singular) probability measure defined on it. I will show that this formalism can account for novel and classical results on the regularity of certain measures.
In connection with the applications of this theory, I will discuss a classical inverse problem in IFS originally introduced by J. H. Elton and Z. Yan in 1987, for which a stable solution algorithm has been derived only recently.