Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle
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,...
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| Format: | Article |
| Language: | English |
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National Research Mordovia State University
2022-02-01
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| Series: | Журнал Средневолжского математического общества |
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| Online Access: | http://journal.svmo.ru/en/archive/article?id=1742 |
| _version_ | 1811336013350436864 |
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| author | Zinina Svetlana Kh. Pochinka Pavel I. |
| author_facet | Zinina Svetlana Kh. Pochinka Pavel I. |
| author_sort | Zinina Svetlana Kh. |
| collection | DOAJ |
| description | 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 $phi'$. |
| first_indexed | 2024-04-13T17:33:09Z |
| format | Article |
| id | doaj.art-3a61caaccfef4560bbec57b63ef57b6d |
| institution | Directory Open Access Journal |
| issn | 2079-6900 2587-7496 |
| language | English |
| last_indexed | 2024-04-13T17:33:09Z |
| publishDate | 2022-02-01 |
| publisher | National Research Mordovia State University |
| record_format | Article |
| series | Журнал Средневолжского математического общества |
| spelling | doaj.art-3a61caaccfef4560bbec57b63ef57b6d2022-12-22T02:37:29ZengNational Research Mordovia State UniversityЖурнал Средневолжского математического общества2079-69002587-74962022-02-01241546510.15507/2079-6900.24.202201.54-65123Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circleZinina Svetlana Kh.0https://orcid.org/0000-0003-3002-281XPochinka Pavel I.1https://orcid.org/0000-0002-6377-747XNational Research Mordovia State University (Saransk, Russian Federation)Higher School of Economics (Nizhny Novgorod, Russian Federation)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 $phi'$.http://journal.svmo.ru/en/archive/article?id=1742rough systems of differential equations”, “rough circle transformations”, “orientation-reversing circle transformations”, “Cartesian product of circle transformations”, “suspension over a diffeomorphism |
| spellingShingle | Zinina Svetlana Kh. Pochinka Pavel I. Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle Журнал Средневолжского математического общества rough systems of differential equations”, “rough circle transformations”, “orientation-reversing circle transformations”, “Cartesian product of circle transformations”, “suspension over a diffeomorphism |
| title | Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle |
| title_full | Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle |
| title_fullStr | Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle |
| title_full_unstemmed | Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle |
| title_short | Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle |
| title_sort | classification of suspensions over cartesian products of orientation reversing diffeomorphisms of a circle |
| topic | rough systems of differential equations”, “rough circle transformations”, “orientation-reversing circle transformations”, “Cartesian product of circle transformations”, “suspension over a diffeomorphism |
| url | http://journal.svmo.ru/en/archive/article?id=1742 |
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