  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.
