ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Golikova I. V., Zinina S. H. Topological conjugacy of n-multiple Cartesian products of circle rough transformations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 6, pp. 851-862. DOI: 10.18500/0869-6632-2021-29-6-851-862

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Article
UDC: 
517.938.5

Topological conjugacy of n-multiple Cartesian products of circle rough transformations

Autors: 
Golikova Iuliana Viktorovna, National Research University "Higher School of Economics"
Zinina Svetlana Halilovna, Mordovia State University
Abstract: 

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

Acknowledgments: 
The study of the dynamics of Cartesian products is supported by the Program “Scientific Foundation of the National Research University Higher School of Economics (HSE)” in 2021–2022 (No. 21-04-004). The classification results were obtained with the support of the RFBR (project 20-31-90069). Also, the authors thank O. V. Pochinka for posing the problem and for useful discussions and E. Y. Gurevich for constructive comments and discussions
Reference: 
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Received: 
28.05.2021
Accepted: 
25.07.2021
Published: 
30.11.2021