The proportion of $k$-cycles for polynomials modulo primes

TL;DR

This paper investigates the distribution of cycle lengths in polynomial iterations over finite fields, establishing bounds on the density of primes where a polynomial exhibits a k-cycle, and providing explicit polynomial families with this property.

Contribution

It proves that for many polynomials, the density of primes with a k-cycle is at most 1/k, and constructs infinite polynomial families with this characteristic.

Findings

Density of primes with a k-cycle is at most 1/k for many polynomials.

Constructs infinite families of polynomials with the property.

Provides bounds on the distribution of cycle lengths in polynomial dynamics over finite fields.

Abstract

Let $f(x) \in \mathbb{F}_p[x]$, and define the orbit of $x\in \mathbb{F}_p$ under the iteration of $f$ to be the set \[ \mathcal{O}(x):=\{x,f(x),(f\circ f)(x),(f\circ f\circ f)(x),\dots\}. \] An orbit is a $k$-cycle if it is periodic of length $k$. In this paper we fix a polynomial $f(x)$ with integer coefficients and for each prime $p$ we consider $f(x) \pmod p$ obtained by reducing the coefficients of $f(x)$ modulo $p$. We ask for the density of primes $p$ such that $f(x)\pmod p$ has a $k$-cycle in $\mathbb{F}_p$. We prove that in many cases the density is at most $1/k$. We also give an infinite family of polynomials in each degree with this property.