Preperiodic points, finiteness, and structures of semigroups of algebraic morphisms

TL;DR

This paper investigates finiteness properties of preperiodic points under algebraic morphisms, connecting group structures with dynamical behavior and extending classical results to new contexts.

Contribution

It introduces new finiteness results for preperiodic points, links group structural properties to preperiodic point commonality, and extends Northcott-type theorems to finite morphisms.

Findings

Finiteness results for preperiodic points under morphisms

Connections between automorphism group properties and preperiodic points

Northcott-type finiteness theorems for finite morphisms

Abstract

In this paper, we explore a variety of finiteness questions for preperiodic points of morphisms. We begin by treating a group action analog of the Burnside problem for torsion groups using the p-adic arc method. We then prove some results connecting commonality of preperiodic points for elements of an automorphism group with structural properties of the group; these results are related to well-known results of Tits and Borel. We finish by proving some Northcott-type results for finite morphisms.