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Russian Journal of Nonlinear Dynamics, 2021, Volume 17, Number 4, Pages 465–473
DOI: https://doi.org/10.20537/nd210408
(Mi nd771)
 

This article is cited in 1 scientific paper (total in 1 paper)

Mathematical problems of nonlinearity

Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

A. I. Morozov

National Research University Higher School of Economics, ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia
Full-text PDF (896 kB) Citations (1)
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Abstract: According to the Nielsen - Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_1^{}$) periodic homeomorphism; $T_2^{}$) reducible non-periodic homeomorphism of algebraically finite order; $T_3^{}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_4^{}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_1^{}$, $T_2^{}$, $T_4^{}$ only. Moreover, all representatives of the class $T_4^{}$ have chaotic dynamics, while in each homotopy class of types $T_1^{}$ and $T_2^{}$ there are regular diffeomorphisms, in particular, Morse - Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse - Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse - Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_1^{}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse - Smale diffeomorphisms belong to types $T_1^{}$ or $T_2^{}$ is uniquely determined by the total intersection index of such knots.
Keywords: Morse – Smale diffeomorphisms, Nielsen – Thurston theory, heteroclinic intersections, homotopy class of a map.
Funding agency Grant number
Russian Science Foundation 21-11-00010
This work was supported by a grant from the Russian Science Foundation, contract 21-11-00010.
Received: 04.12.2021
Accepted: 13.12.2021
Bibliographic databases:
Document Type: Article
MSC: 37D05
Language: english
Citation: A. I. Morozov, “Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection”, Rus. J. Nonlin. Dyn., 17:4 (2021), 465–473
Citation in format AMSBIB
\Bibitem{Mor21}
\by A. I. Morozov
\paper Determination of the Homotopy Type of
a Morse – Smale Diffeomorphism on a 2-torus
by Heteroclinic Intersection
\jour Rus. J. Nonlin. Dyn.
\yr 2021
\vol 17
\issue 4
\pages 465--473
\mathnet{http://mi.mathnet.ru/nd771}
\crossref{https://doi.org/10.20537/nd210408}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85123525573}
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