Известия высших учебных заведений: Прикладная нелинейная динамика (Nov 2021)

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

  • Golikova, Iuliana Viktorovna,
  • Zinina, Svetlana Halilovna

DOI
https://doi.org/10.18500/0869-6632-2021-29-6-851-862
Journal volume & issue
Vol. 29, no. 6
pp. 851 – 862

Abstract

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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.

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