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This article is cited in 1 scientific paper (total in 1 paper)
Mathematics
On perturbations of algebraic periodic automorphisms of a two-dimensional torus
V. Z. Grines, D. I. Mints, E. E. Chilina National Research University – Higher School of Economics in Nizhny Novgorod
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 two-dimensional 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.
Keywords:
two-dimensional torus, nonhyperbolic algebraic automorphism, one-parameter families.
Citation:
V. Z. Grines, D. I. Mints, E. E. Chilina, “On perturbations of algebraic periodic automorphisms of a two-dimensional torus”, Zhurnal SVMO, 24:2 (2022), 141–150
Linking options:
https://www.mathnet.ru/eng/svmo825 https://www.mathnet.ru/eng/svmo/v24/i2/p141
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Abstract page: | 106 | Full-text PDF : | 43 | References: | 24 |
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