Periodic subvarieties of semiabelian varieties and annihilators of irreducible representations

TL;DR

This paper proves that semiabelian varieties have no proper subvarieties intersecting all periodic point orbits under an endomorphism, and applies this to characterize annihilator ideals in skew polynomial algebras.

Contribution

It establishes a new non-existence result for subvarieties intersecting all periodic orbits and links this to the structure of annihilator ideals in skew polynomial algebras.

Findings

No proper subvariety intersects all periodic point orbits.

Topological characterization of annihilator ideals in skew polynomial algebras.

Extension of dynamics results to algebraic and noncommutative settings.

Abstract

Let $G$ be a semiabelian variety defined over a field of characteristic $0$, endowed with an endomorphism $\Phi$. We prove there is no proper subvariety $Y\subset G$ which intersects the orbit of each periodic point of $G$ under the action of $\Phi$. As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras.