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.1991 Mathematics Subject Classification. 28A80 (primary), 37H99, 60G57, 60J05 (secondary).

Abstract. We describe new families of random fractals, referred to as "Vvariable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V -variable fractals has at most V key "forms" or "shapes". V -variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V -variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example R 2 , and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

The families of V -variable fractals for V D 1; 2; 3; : : : , together with their natural probability distributions, interpolate between the corresponding families of random homogeneous fractals and of random recursive fractals. We investigate certain random V V matrices associated with these fractals and use them to compute the almost sure Hausdorff dimension of V -variable fractals satisfying the uniform open set condition.

We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.Comment: 22 page

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