Periods of iterations of functions with restricted preimage sizes
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.
Keywords
Cite
@article{arxiv.1701.09148,
title = {Periods of iterations of functions with restricted preimage sizes},
author = {Rodrigo S. V. Martins and Daniel Panario and Claudio Qureshi and Eric Schmutz},
journal= {arXiv preprint arXiv:1701.09148},
year = {2018}
}
Comments
An extended abstract of this work was presented at the 29th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Some corrections and improvevments have been made