Preperiodic points of polynomial dynamical systems over finite fields

TL;DR

This paper studies the distribution of preperiodic points in polynomial dynamical systems over finite fields, providing bounds on their proportions and generalizations for broader classes of polynomials.

Contribution

It introduces effective bounds on the proportion of preperiodic points for unicritical polynomials over finite fields and extends these results to generalized definitions.

Findings

Derived upper bounds for the proportion of preperiodic points.

Established bounds for generalized preperiodic point sets.

Analyzed the distribution of preperiodic points in finite field dynamical systems.

Abstract

For a prime $p$, positive integers $r,n$, and a polynomial $f$ with coefficients in $\mathbb{F}_{p^r}$, let $W_{p,r,n}(f)=f^n\left(\mathbb{F}_{p^r}\right)\setminus f^{n+1}\left(\mathbb{F}_{p^r}\right)$. As $n$ varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of $f$ on $\mathbb{F}_{p^r}$. In this paper we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of $\mathbb{F}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages.