Periods of Iterated Rational Functions over a Finite Field

TL;DR

This paper investigates the behavior of iterated rational functions over finite fields, estimating their periods and analyzing the distribution of cycle lengths in the associated functional graph.

Contribution

It provides new estimates for the ultimate periods of iterated rational functions and characterizes the joint distribution of small cycle lengths in their functional graphs.

Findings

Estimated the ultimate period of random rational functions over finite fields.

Determined the joint distribution of small cycle lengths in the associated graph.

Used Lagrange interpolation and factorial moments in proofs.

Abstract

Choose a random degree d poly f with coefficients in a finite field F. We estimate the ultimate period of f under compositional iteration. We also determine the joint distribution of the small cycle lengths in the graph with edges (x,f(x)), x in F. The proofs use Lagrange interpolation and the method of factorial moments.