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.
"Nonuniform random transformations." Ann. Appl. Probab. 10 (4) 1151 - 1181, November 2000. https://doi.org/10.1214/aoap/1019487611