On preperiodic points of rational functions defined over $\mathbb{F}_p(t)$

TL;DR

This paper investigates the behavior of preperiodic points of rational functions over the function field _p(t), establishing bounds on the size of their orbits and extending previous results from number fields to function fields.

Contribution

It generalizes known results about periodic points from number fields to rational functions over _p(t) with good reduction everywhere.

Findings

Bounds on the cardinality of orbits for periodic points.

Extension of results from number fields to function fields.

Analysis of preperiodic points over _p(t).

Abstract

Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of this fact to the set of all rational functions defined over ${\mathbb{Q}}$ with good reduction everywhere (i.e. at any finite place of $\mathbb{Q}$). The set of monic polynomials with coefficients in $\mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2({\mathbb{Z}})$, as the set of all rational functions defined over $\mathbb{Q}$ with a totally ramified fixed point in $\mathbb{Q}$ and with good reduction everywhere. Let $p$ be a prime number and let ${\mathbb{F}}_p$ be the field with $p$ elements. In the present paper we consider rational functions defined over the rational global function field ${\mathbb{F}}_p(t)$ with good reduction at every finite place. We prove some bounds for the cardinality of orbits in ${\mathbb{F}}_p(t)\cup \{\infty\}$ for periodic and preperiodic points.