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:
T1) 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 T1, T2, T4 only. Moreover, all representatives of the class T_4 have chaotic dynamics, while in each homotopy class of types T1 and T2 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 T1. 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 T1 or T2 is uniquely determined by the total intersection index of such knots.