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

V. Z. Grines ; A. I. Morozov ; O. V. Pochinka

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 95-107

Résumé

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.