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

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.