Period sets of linear toral endomorphisms on $T^2$

TL;DR

This paper classifies the possible sets of periods for all linear endomorphisms of the 2D torus, extending known results to include non-invertible maps and providing a complete characterization.

Contribution

It provides a comprehensive classification of period sets for linear toral endomorphisms on ^2, including non-invertible cases, based on homotopy class intersections.

Findings

Complete classification of period sets for 2D toral endomorphisms.

Extension of known results to non-invertible maps.

Identification of conditions for specific period sets.

Abstract

The period set of a dynamical system is defined as the subset of all integers $n$ such that the system has a periodic orbit of length $n$. Based on known results on the intersection of period sets of torus maps within a homotopy class, we give a complete classification of the period sets of (not necessarily invertible) toral endomorphisms on the $2$--dimensional torus $\mathbb{T}^2$.