ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.24.202201.54-65

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 57N10

Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle

S. K. Zinina1, P. I. Pochinka2

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

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

Abstract. This paper introduces class $G$ containing Cartesian products of orientation-changing rough transformations of the circle and studies their dynamics. As it is known from the paper of A. G. Maier non-wandering set of orientation-changing diffeomorphism of the circle consists of $2q$ periodic points, where $q$ is some natural number. So Cartesian products of two such diffeomorphisms has $4q_1q_2$ periodic points where $q_1$ corresponds to the first transformation and $q_2$ corresponds to the second one. The authors describe all possible types of the set of periodic points, which contains $2q_1q_2$ saddle points, $q_1q_2$ sinks, and $q_1q_2$ sources; $4$ points from mentioned $4q_1q_2$ periodic ones are fixed, and the remaining $4q_1q_2-4$ points have period $2$. In the theory of smooth dynamical systems, a very useful result is that, given a diffeomorphism $f$ of a manifold, one can construct a flow on a manifold with dimension one greater; this flow is called the suspension over $f$. The authors introduce the concept of suspension over diffeomorphisms of class $G$, describe all possible types of suspension orbits and the number of these orbits. Besides that, the authors prove a theorem on the topology of the manifold on which the suspension is given. Namely, the carrier manifold of the flows under consideration is homeomorphic to the closed 3-manifold $\mathbb T^2 \times [0,1]/\varphi$, where $\varphi :\mathbb T^ 2 \to \mathbb T^2$. The main result of the paper says that suspensions over diffeomorphisms of the class $G$ are topologically equivalent if and only if corresponding diffeomorphisms are topologically conjugate. The idea of the proof is to show that the topological equivalence of the suspensions $\phi^t$ and $\phi'^t$ implies the topological conjugacy of $\phi$ and $\phi'$.

Key Words: rough systems of differential equations, rough circle transformations, orientation-reversing circle transformations, Cartesian product of circle transformations, suspension over a diffeomorphism

For citation: S. K. Zinina, P. I. Pochinka. Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:1(2022), 54–65. DOI:

Submitted: 01.12.2021; Revised: 10.02.2022; Accepted: 24.02.2022

Information about the authors:

Svetlana Kh. Zinina, Postgraduate Student, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID:,

Pavel I. Pochinka, Student of the Faculty of Informatics, Mathematics and Computer Science, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID:,

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.