A hyperbolic IteratedFunctionSystem with probablities {X: f1,f2,...,fN;p1,p2,...,pN} is a hyperbolic IFS {X:f1,f2,...,fN}, with X a compact metric space, s.t. each fi (i=1,2,...,N) is assigned a probability pi with 0<pi<1 and sum(i=1toN) pi=1.

A hyperbolic IFS with probabilities induces a ContractingMap on the set of normalized BorelMeasures MX on X with a unique fixed point described as a MultiFractal whose support is the attractor of {X:f1,f2,...,fN}.

In the extreme case where all the maps fi(i=1,2,...,N) are the same, the unique invariant measure has the DiracDeltaFunction as its density and its support is a single point corresponding to the unique fixed point of the map.

And beyond that, you need the article: ["ASmoothApproximationOnTheEdgeOfChaos"]