Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms

TL;DR

This paper investigates the structure of periodic and eventually periodic points in affine infra-nilmanifold endomorphisms, providing conditions for their existence and density, and fully characterizing these points.

Contribution

It offers new criteria for (eventually) periodic points and describes their distribution, advancing understanding of dynamical behavior on infra-nilmanifolds.

Findings

Periodic points form a dense subset if they exist.

Necessary conditions for eventually periodic points are established.

Complete description of eventually periodic points for any affine infra-nilmanifold endomorphism.

Abstract

In this paper, we study the periodic and eventually periodic points of affine infra-nilmanifold endomorphisms. On the one hand, we give a sufficient condition for a point of the infra-nilmanifold to be (eventually) periodic. In this way we show that if an affine infra-nilmanifold endomorphism has a periodic point, then its set of periodic points forms a dense subset of the manifold. On the other hand, we deduce a necessary condition for eventually periodic points from which a full description of the set of eventually periodic points follows for an arbitrary affine infra-nilmanifold endomorphism.