On non-hyperbolic algebraic automorphisms of a two-dimensional torus

Sergey V Sidorov,Ekaterina E Chilina

doi:10.15507/2079-6900.23.202103.295-307

Abstract

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, 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. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

Highlights

This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S
Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V
Bezdenezhykh, 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

Summary

Introduction

Если собственные значения матрицы A ∈ Gl(2, Z) не равны по модулю единице, то алгебраический автоморфизм двумерного тора, индуцированный матрицей A, называется гиперболическим алгебраическим автоморфизмом двумерного тора. Два алгебраических автоморфизма тора f и g называются сопряженными, если существует такой автоморфизм h, что g = hf h−1. Задача о нахождении классов сопряженности негиперболических алгебраических автоморфизмов двумерного тора сводится к задаче о нахождении классов подобия целочисленных унимодулярных матриц второго порядка, собственные значения которых равны по модулю единице.

Results

Conclusion