Preperiodic points for rational functions defined over a rational function field of characteristic zero

TL;DR

This paper establishes a bound on the number of rational preperiodic points for non-isotrivial rational functions over a rational function field of characteristic zero, linking it to places of bad reduction and the function's degree.

Contribution

It provides a new explicit bound on preperiodic points for rational functions over function fields, extending understanding of dynamical systems in this setting.

Findings

Bound depends on places of bad reduction and degree d

Finite set of preperiodic points under given conditions

Advances in arithmetic dynamics over function fields

Abstract

Let $k$ be an algebraic closed field of characteristic zero. Let $K$ be the rational function field $K=k(t)$. Let $\phi$ be a non isotrivial rational function in $K(z)$. We prove a bound for the cardinality of the set of $K$--rational preperiodic points for $\phi$ in terms of the number of places of bad reduction and the degree $d$ of $\phi$.