Andrei Morozov
Bolshaya Pecherskaya street, 25/12, 603155
National Research University Higher School of Econ
Publications:
Morozov A. I.
Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2torus by Heteroclinic Intersection
2021, Vol. 17, no. 4, pp. 465473
Abstract
According to the Nielsen – Thurston classification, the set of homotopy classes of orientationpreserving 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 nonperiodic homeomorphism of algebraically finite order; $T_3^{}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_4^{}$) pseudoAnosov 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 twodimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and nontrivial. 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 nontrivial heteroclinic domains consists of a finite number of twodimensional 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.
