Upper Bounds on the Sizes of Finite Orbits for Unramified Morphisms

TL;DR

This paper establishes effective upper bounds for the size of finite orbits in algebraic dynamical systems with unramified morphisms, enabling decidability of whether a point has a finite orbit.

Contribution

It provides the first effective bounds for finite orbits in systems generated by unramified endomorphisms over algebraic varieties.

Findings

Finite orbits have bounded size in these systems.

Decidability of finite orbit existence is achieved.

Effective bounds depend on the system's properties.

Abstract

We prove that for a dynamical system on an algebraic variety over $\overline{\mathbb{Q}}$ generated by finitely many unramified endomorphisms, it is decidable whether a given point has a finite orbit. This is achieved by establishing an effective upper bound for the size of finite orbits in integral algebraic dynamical systems with unramified morphisms.