V-Variable Fractals: Fractals with Partial Self Similarity

TL;DR

This paper introduces V-variable fractals, a new class of fractals with partial self-similarity at all scales, extending classical IFS theory and providing foundational properties and approximation results.

Contribution

It defines V-variable fractals, proves their existence and uniqueness, and extends classical IFS results to this new framework.

Findings

V-variable fractals exist for any collection of IFS and probability distribution.

They can be approximated from deterministic IFS attractors.

Basic properties of classical IFS are extended to V-variable fractals.

Abstract

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. 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.