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Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 2022, Volume 24, Number 2, Pages 141–150
DOI: https://doi.org/10.15507/2079-6900.24.202202.141-150
(Mi svmo825)
 

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
Full-text PDF (525 kB) Citations (1)
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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.
Funding agency Grant number
HSE Academic Fund Programme 21-04-004
National Research University Higher School of Economics 075-15-2019-1931.
Document Type: Article
UDC: 517.938
MSC: 37C05
Language: Russian
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
Citation in format AMSBIB
\Bibitem{GriMinChi22}
\by V.~Z.~Grines, D.~I.~Mints, E.~E.~Chilina
\paper On perturbations of algebraic periodic automorphisms of a two-dimensional torus
\jour Zhurnal SVMO
\yr 2022
\vol 24
\issue 2
\pages 141--150
\mathnet{http://mi.mathnet.ru/svmo825}
\crossref{https://doi.org/10.15507/2079-6900.24.202202.141-150}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva
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