  This paper studies a general class of Iterated Function Systems (IFS). No
contractivity assumptions are made, other than the existence of some compact
attractor. The possibility of escape to infinity is considered. Our present
approach is based on Hilbert space, and the theory of representations of the
Cuntz algebras O_n, n=2,3,.... While the more traditional approaches to IFS's
start with some equilibrium measure, ours doesn't. Rather, we construct a
Hilbert space directly from a given IFS; and our construction uses instead
families of measures. Starting with a fixed IFS S_n, with n branches, we prove
existence of an associated representation of O_n, and we show that the
representation is universal in a certain sense. We further prove a theorem
about a direct correspondence between a given system S_n, and an associated
sub-representation of the universal representation of O_n.
