DOI 10.15507/2079-6900.24.202202.141-150

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 37C05

On perturbations of algebraic periodic automorphisms of a two-dimensional torus

V. Z. Grines, D. I. Mints, E. E. Chilina

Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. According to the results of V. Z. Grines and A. N. Bezdenezhnykh, for each gradient-like diffeomorphism of a closed orientable surface $M^2$ there exist a gradient-like flow and a periodic diffeomorphism of this surface such that the original diffeomorphism is a superposition of a diffeomorphism that is a shift per unit time of the flow and the periodic diffeomorphism. In the case when $M^2$ is a two-dimensional torus, there is a topological classification of periodic maps. Moreover, it is known that there is only a finite number of topological conjugacy classes of periodic diffeomorphisms that are not homotopic to identity one. Each such class contains a representative that is a periodic algebraic automorphism of a two-dimensional torus. Periodic automorphisms of a two-dimensional torus are not structurally stable maps, and, in general, it is impossible to predict the dynamics of their arbitrarily small perturbations. However, in the case when a periodic diffeomorphism is algebraic, we constructed a one-parameter family of maps consisting of the initial periodic algebraic automorphism at zero parameter value and gradient-like diffeomorphisms of a twodimensional torus for all non-zero parameter values. Each diffeomorphism of the constructed one-parameter families inherits, in a certain sense, the dynamics of a periodic algebraic automorphism being perturbed.

Key Words: two-dimensional torus, nonhyperbolic algebraic automorphism, one-parameter families

For citation: V. Z. Grines, D. I. Mints, E. E. Chilina. On perturbations of algebraic periodic automorphisms of a two-dimensional torus. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:2(2022), 141–150. DOI: https://doi.org/10.15507/2079-6900.24.202202.141-150

Submitted: 30.03.2021; Revised: 02.05.2022; Accepted: 25.05.2022

Information about the authors:

Vyacheslav Z. Grines, Professor of the Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603150, Russia), Dr.Sci. (Phys.-Math.), ORCID: https://orcid.org/0000-0003-4709-6858, vgrines@yandex.ru

Dmitrii I. Mints, Research Assistant, International Laboratory of Dynamical Systems and Applications, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: https://orcid.org/0000-0003-0329-6946, dmitriimints@gmail.com

Ekaterina E. Chilina, Research Assistant, International Laboratory of Dynamical Systems and Applications, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: https://orcid.org/0000-0002-1298-9237, k.chilina@yandex.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.