On roots of unity in orbits of rational functions

TL;DR

This paper characterizes rational functions over number fields whose orbits contain infinitely many roots of unity within the cyclotomic closure, extending previous work on polynomials with many preperiodic points.

Contribution

It provides a complete characterization of such rational functions, generalizing known results from polynomial cases to rational functions.

Findings

Identifies conditions for rational functions with infinitely many roots of unity in orbits

Extends results from polynomial to rational function dynamics

Connects orbit properties with cyclotomic field extensions

Abstract

In this paper we characterise univariate rational functions over a number field $\K$ having infinitely many points in the cyclotomic closure $\K^c$ for which the orbit contains a root of unity. Our results are similar to previous results of Dvornicich and Zannier describing all polynomials having infinitely many preperiodic points in $\K^c$.