Geometrically Nilpotent Subvarieties

TL;DR

This paper constructs examples of polynomial maps over finite fields with subvarieties where each point eventually maps to a fixed point, but the subvariety as a whole does not, raising interesting questions in algebraic dynamics.

Contribution

It introduces specific polynomial maps with unique dynamical properties on subvarieties over finite fields, highlighting new phenomena in algebraic dynamics.

Findings

Existence of polynomial maps with special subvarieties

Subvarieties where points are eventually fixed but the whole is not

Open questions in algebraic dynamics

Abstract

We construct some examples of polynomial maps over finite fields that admit subvarieties with a peculiar property: every geometric point is mapped to a fixed point by some iteration of the map, while the whole subvariety is not. Several related open questions are stated and discussed.