Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection

According to Thurston's classification, the set of homotopy classes of homeomorphisms defined on closed orientable surfaces of negative curvature is split into four disjoint subsets $T_1$, $T_2$, $T_3$, and $T_4$. A homotopy class from each subset is characterized by the existence in it of a homeomorphism (called the Thurston canonical form) that is exactly of one of the following types, respectively: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of algebraically finite order, or a pseudo-Anosov homeomorphism. Thurston's canonical forms are not structurally stable diffeomorphisms. Therefore, the problem of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class arises naturally. A. N. Bezdenezhnykh and V. Z. Grines constructed a gradient-like diffeomorphism in each homotopy class from $T_1$. R. V. Plykin and A. Yu. Zhirov announced a method for constructing a structurally stable diffeomorphism in each homotopy class from $T_4$. The nonwandering set of this diffeomorphism consists of a finite number of source orbits and a single one-dimensional attractor. In the present paper, we describe the construction of a structurally stable diffeomorphism in each homotopy class from $T_2$. The constructed representative is a Morse–Smale diffeomorphism with an orientable heteroclinic intersection.