Dimension and measure for typical random fractals

TL;DR

This paper investigates the typical Hausdorff and packing dimensions and measures of attractors generated by random iterated function systems, revealing contrasting results to probabilistic approaches and demonstrating complex phenomena without separation assumptions.

Contribution

It computes the Baire-typical dimensions of random attractors in RIFS, contrasting with almost sure results, and explores measure properties under minimal assumptions.

Findings

Typical dimensions differ from probabilistic results.

Examples show complex measure phenomena.

Results hold without separation conditions.

Abstract

We consider the dimension and measure of typical attractors of random iterated function systems (RIFSs). We define a RIFS to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space and, for a given RIFS, we define a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the 'almost sure' dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give a number of examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.