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

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 $f(x)(\mathrm{mod}p)$ obtained by reducing the coefficients of $f(x)$ modulo $p$. We ask for the density of primes $p$ such that $f(x)(\mathrm{mod}p)$ 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.