Periods of iterated rational functions

Abstract

If [Formula: see text] is a polynomial of degree [Formula: see text] in [Formula: see text], let [Formula: see text] be the number of cycles of length [Formula: see text] in the directed graph on [Formula: see text] with edges [Formula: see text] For random polynomials, the numbers [Formula: see text] have asymptotic behavior resembling that for the cycle lengths of random functions [Formula: see text] However random polynomials differ from random functions in important ways. For example, given the set of cyclic (periodic) points, it is not necessarily true that all permutations of those cyclic points are equally likely to occur as the restriction of [Formula: see text]. This, and the limitations of Lagrange interpolation, together complicate research on [Formula: see text] the ultimate period of [Formula: see text] under compositional iteration. We prove a lower bound for the average value of [Formula: see text]: if [Formula: see text], but [Formula: see text], then the expected value of [Formula: see text] is [Formula: see text] where the sum is over all [Formula: see text] polynomials of degree [Formula: see text] in [Formula: see text]. Similar results are proved for rational functions.