Realization of Morse-Smale Surface Diffeomorphisms with Oriented Heteroclinic

In this paper, we consider a class of orientation-preserving Morse-Smale diffeomorphisms defined on an orientable surface. In papers by A.A. Bezdenezhnykh and V.Z. Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of a heteroclinic intersection. However, such graphs generally do not admit polynomial discriminating algorithms. The main result of my thesis is a part of the classification, namely, a class of admissible graphs has been distinguished, each class of isomorphism of which can be realized by a diffeomorphism of a surface with an orientable heteroclinic. The complete classification is described in a recent paper by D.S. Malishev, A.I.Morozov and O.V. Pochinka. The results obtained are directly related to the realization problem of homotopy classes of homeomorphisms on closed orientable surfaces. In particular, this classification give an approach to constructing a representative in each homotopy class of homeomorphisms of algebraically finite type according to the Nielsen classification, which is an open problem today.