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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
Аннотация:
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.
Ключевые слова:
Morse – Smale diffeomorphisms, Nielsen – Thurston theory, heteroclinic intersections,
homotopy class of a map.
MSC: 37D05 Поступила в редакцию: 04.12.2021
Принята в печать: 13.12.2021
Язык публикации: английский
DOI:
10.20537/nd210408