Geodesic
Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection
Informatics and Automation, Optimal Control and Differential Games, Tome 315 (2021), pp. 95-107 
V. Z. Grines; A. I. Morozov; O. V. Pochinka. Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection. Informatics and Automation, Optimal Control and Differential Games, Tome 315 (2021), pp. 95-107. http://geodesic.mathdoc.fr/item/TRSPY_2021_315_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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