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

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.Comment: arXiv admin note: substantial text overlap with arXiv:1403.229