Collisions and incidence of vertices and components in the graph of $k$-fold iteration of the uniform random mapping

The probabilistic characteristics of the graph of $k$-fold iteration of uniform random mapping are studied. Formulas for the distribution of the length of the aperiodicity segment of an arbitrary vertex with some restrictions are calculated. We obtain exact expressions for the probabilities that two arbitrary vertices belong to the same connected component, that an arbitrary vertex belongs to the preimage set of another vertex and that there exists a collision in the considered graph.} \keywords{ uniform random mapping, iteration of random mapping, aperiodicity segment, graph of a mapping, connected component, preimage, collision