On Endomorphisms of Projective Algebraic Varieties

TL;DR

This paper investigates the algebraic dynamics of endomorphisms on projective varieties, focusing on their iterated images, Stein factorizations, and stability phenomena, extending previous results in the field.

Contribution

It provides new characterizations of iterated images, analyzes Stein factorizations for stability, and completes a classification of endomorphisms with iterates in finite unions of components.

Findings

Characterized the intersection of images of iterates of endomorphisms.

Proved stability phenomena in Stein factorizations.

Extended Brion's result on endomorphisms with iterates in finite unions.

Abstract

We study the algebraic dynamics of endomorphisms of projective varieties. First, we characterize their iterated images, i.e. the intersection of the images of their iterates. Next, we explore the Stein factorizations of the iterates, proving some stability phenomena they exhibit. Finally, we study endomorphisms whose iterates lie in a finite union of connected components of the endomorphism scheme, thereby completing a result of Brion.