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

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