Geodesic
Topological conjugacy of n-multiple Cartesian products of circle rough transformations
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 29 (2021) no. 6, pp. 851-862.

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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.
Keywords: Morse - Smale diffeomorphisms, circle rough transformations, rotation number, periodic orbits, topological invariants. 
@article{IVP_2021_29_6_a2,
     author = {I. V. Golikova and S. Kh. Zinina},
     title = {Topological conjugacy of n-multiple {Cartesian} products of circle rough transformations},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {851--862},
     publisher = {mathdoc},
     volume = {29},
     number = {6},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2021_29_6_a2/}
}
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I. V. Golikova; S. Kh. Zinina. Topological conjugacy of n-multiple Cartesian products of circle rough transformations. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 29 (2021) no. 6, pp. 851-862. http://geodesic.mathdoc.fr/item/IVP_2021_29_6_a2/