Scarcity of cycles for rational functions over a number field

TL;DR

This paper establishes explicit bounds on the number of periodic points for rational functions over number fields, depending on primes of bad reduction and degree, with improvements under certain conditions.

Contribution

It provides the first explicit bounds on periodic points for rational functions over number fields, depending only on specific invariants and extends to finitely generated semigroups.

Findings

Bound depends linearly on the degree of the rational function.

Explicit bounds depend only on primes of bad reduction and degree.

Under stronger assumptions, the degree dependence can be eliminated.

Abstract

We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the degree. More generally, we show that there exists an explicit uniform bound on the number of periodic points for any rational function in a given finitely generated semigroup (under composition) of rational functions of degree at least 2. We show that under stronger assumptions the dependence on the degree of the map in the bounds can be removed.