With a given transformation on a finite domain, we associate a
three-dimensional distribution function describing the component size, cycle
length and trajectory length of each point in the domain.We then consider a
random transformation on the domain, in which images of points are independent
and identically distributed. The three-dimensional distribution function
associated with this random transformation is itself random. We show that,
under a simple homogeneity condition on the distribution of images, and with a
suitable scaling, this random distribution function has a limit law as the
number of points in the domain tends to $\infty$. The proof is based on a
Poisson approximation technique for matches in an urn model. The result helps
to explain the behavior of computer implementations of chaotic dynamical
systems.