# Periods of iterations of functions with restricted preimage sizes

## TL;DR

This paper analyzes the asymptotic behavior of cycle-related properties in random mappings with restricted preimage sizes, providing new insights into their distribution and expectations, with applications to polynomial models over finite fields.

## Contribution

It introduces the asymptotic lognormality of cycle length products and least common multiples in restricted mappings, extending understanding of their probabilistic structure.

## Key findings

- Both T and B are asymptotically lognormal.
- Expected values of T and B are estimated and validated numerically.
- Results have implications for polynomial models over finite fields.

## Abstract

We consider random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) is greater than 1. We prove that T, the least common multiple of the cycle lengths, and B= the product of the cycle lengths, are both asymptotically lognormal. The expected values of these random variables are also also estimated and compared with numerical results. This work is motivated, in part, by the use of these mappings as heuristic models for polynomials of the form x^k + a over the integers modulo p with p congruent to 1 mod k.