S. V. Sidorov¹, E. E. Chilina²

¹National Research Lobachevsky State University of Nizhny Novgorod (Nizhny Novgorod, Russian Federation)

²National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

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.

Key Words: periodic homeomorphisms, two-dimensional torus, algebraic automorphism

For citation: S. V. Sidorov, E. E. Chilina. On non-hyperbolic algebraic automorphisms of a two-dimensional torus. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:3(2021), 295–307. DOI: https://doi.org/10.15507/2079-6900.23.202103.295–307

Information about the authors:

Sergey V. Sidorov, Associate Professor, Department of Algebra, Geometry and Discrete Mathematics, National Research Lobachevsky State University of Nizhny Novgorod (23 Gagarina Av., Nizhny Novgorod 603950, Russia), Ph. D. (Physics and Mathematics), ORCID: https://orcid.org/0000-0003-2883-6427, sesidorov@yandex.ru

Ekaterina E. Chilina, student of National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), k.chilina@yandex.ru