S. K. Zinina¹, A. A. Nozdrinov², V. I. Shmukler²

¹National Research Mordovia State University (Saransk, Russian Federation)

²Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. One of the constructions for obtaining flows on a manifold is building a superstructure over a cascade. In this case, the flow is non-singular, that is, it has no fixed points. C. Smale showed that superstructures over conjugate diffeomorphisms are topologically equivalent. The converse statement is not generally true, but under certain assumptions the conjugacy of diffeomorphisms is tantamount to equivalence of superstructures. Thus, J. Ikegami showed that the criterion works in the case when a diffeomorphism is given on a manifold whose fundamental group does not admit an epimorphism into the group $\mathbb Z$. He also constructed examples of non-conjugate diffeomorphisms of a circle whose superstructures are equivalent. In the work of I. V. Golikova and O. V. Pochinka superstructures over diffeomorphisms of circles are examined. It is also proven in this paper that the complete invariant of the equivalence of superstructures over orientation-preserving diffeomorphisms is the equality of periods for periodic points generating their diffeomorphisms. For the other side, it is known from the result of A.G. Mayer that the coincidence of rotation numbers is also necessary for conjugacy of orientation-preserving diffeomorphisms. At the same time, superstructures over orientation-changing diffeomorphisms of circles are equivalent if and only if the corresponding diffeomorphisms of circles are topologically conjugate. Work of S. Kh. Zinina and P. I. Pochinka proved that superstructures over orientation-changing Cartesian products of diffeomorphisms of circles are equivalent if and only if the corresponding diffeomorphisms of tori are topologically conjugate. In this paper a classification result is obtained for superstructures over Cartesian products of orientation-preserving diffeomorphisms of circles.

Key Words: manifold, superstructure over a diffeomorphism, orientation-preserving diffeomorphism of a circle, number of rotations, Cartesian product of diffeomorphisms

For citation: S. K. Zinina, A. A. Nozdrinov, V. I. Shmukler. Superstructures over Cartesian products of orientation-preserving rough circle transformations. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:4(2023), 273–283. DOI: https://doi.org/10.15507/2079-6900.25.202304.273-283

Information about the authors:

Svetlana Kh. Zinina, Senior Lecturer, Department of Mathematical Analysis, Algebra and Geometry, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Math.), ORCID: http://orcid.org/0000-0003-3002-281X,zininaskh@math.mrsu.ru

Alexey A. Nozdrinov, Post-graduate student, Intern researcher at the Laboratory of Dynamic Systems and Applications, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0002-1223-7334

Valeria I. Shmukler, Post-graduate student, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0003-3125-1825, shmukler9797@mail.ru