On classification of periodic maps on the 2-torus

TL;DR

This paper classifies all orientation-preserving periodic maps on the 2-torus, providing a complete characteristic, realizing all classes, and linking to the topological classification of gradient-like surface diffeomorphisms.

Contribution

It introduces a complete characteristic for such maps, realizes all admissible classes, and connects these results to the classification of gradient-like surface diffeomorphisms.

Findings

All admissible classes of periodic maps are realized.

Number of non-homotopic classes is finite.

Each class corresponds to an algebraic automorphism.

Abstract

In this paper, following J.Nielsen, we introduce a complete characteristic of orientation preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of classes of non-homotopic to the identity orientation preserving periodic homeomorphisms on the 2-torus is realized by an algebraic automorphism. Moreover, it is shown that number of such classes is finite. Due to V.Z. Grines and A. Bezdenezhnykh, any gradient like orientation preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. So, results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.