MSC2010 05C62, 14J80, 37D15
Classification of rough transformations of a circle from a modern point of view
A. E. Kolobyanina1, E. V. Nozdrinova2, O. V. Pochinka3
| Annotation | In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known. |
|---|---|
| Keywords | rough transformations of a circle, topological classification |
1Anna E. Kolobyanina, Student of the educational program "Mathematics", National Research University Higher School of Economics (25/12, Bolshaya Pecherskaya St., Nizhniy Novgorod, 603155, Russia), ORCID: http://orcid.org/0000-0001-5312-4478, anyakolobyanina@mail.ru
2Elena V. Nozdrinova, Trainee Researcher, Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics (Bolshaya Pecherskaya St., Nizhniy Novgorod, 603155, Russia), ORCID: http://orcid.org/0000-0001-5209-377X, maati@mail.ru
3Olga V. Pochinka, Laboratory Head, Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics (Bolshaya Pecherskaya St., Nizhniy Novgorod, 603155, Russia), ORCID: http://orcid.org/0000-0002-6587-5305,olga-pochinka@yandex.ru
Citation: A. E. Kolobyanina, E. V. Nozdrinova, O. V. Pochinka, "[Classification of rough transformations of a circle from a modern point of view]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:4 (2018) 408–418 (In Russian)
DOI 10.15507/2079-6900.20.201804.408-418
