Periods of Iterations of Functions with Restricted Preimage Sizes

Abstract

Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.