V -variable Fractals: Fractals with Partial Self Similarity

We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V -variable fractal sets or measures with a natural probability distribution. These V -variable fractals can be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems.