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: https://doi.org/10.15507/2079-6900.24.202201.54-65

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: http://orcid.org/0000-0003-3002-281X, kapkaevasvetlana@yandex.ru

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: https://orcid.org/0000-0002-6377-747X, pavel-pochinka@yandex.ru