Geodesic
Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 95-107 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{TM_2021_315_a6,
     author = {V. Z. Grines and A. I. Morozov and O. V. Pochinka},
     title = {Realization of {Homeomorphisms} of {Surfaces} of {Algebraically} {Finite} {Order} by {Morse{\textendash}Smale} {Diffeomorphisms} with {Orientable} {Heteroclinic} {Intersection}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {95--107},
     year = {2021},
     volume = {315},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_315_a6/}
}
TY  - JOUR
AU  - V. Z. Grines
AU  - A. I. Morozov
AU  - O. V. Pochinka
TI  - Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2021
SP  - 95
EP  - 107
VL  - 315
UR  - http://geodesic.mathdoc.fr/item/TM_2021_315_a6/
LA  - ru
ID  - TM_2021_315_a6
ER  - 
%0 Journal Article
%A V. Z. Grines
%A A. I. Morozov
%A O. V. Pochinka
%T Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2021
%P 95-107
%V 315
%U http://geodesic.mathdoc.fr/item/TM_2021_315_a6/
%G ru
%F TM_2021_315_a6
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. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 95-107. http://geodesic.mathdoc.fr/item/TM_2021_315_a6/

[1] Abikoff W., “The uniformization theorem”, Amer. Math. Mon., 88:8 (1981), 574–592 | DOI

[2] S. Kh. Aranson and V. Z. Grines, “The topological classification of cascades on closed two-dimensional manifolds”, Russ. Math. Surv., 45:1 (1990), 1–35 | DOI | MR | MR

[3] Bezdenezhnykh A.N., Topologicheskaya klassifikatsiya diffeomorfizmov Morsa–Smeila s orientiruemym geteroklinicheskim mnozhestvom na dvumernykh mnogoobraziyakh, Dis. ... kand. fiz.-mat. nauk, Gork. gos. un-t, Gorkii, 1985

[4] A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I”, Sel. Math. Sov., 11:1 (1992), 1–11 | MR | MR

[5] Bezdenezhnykh A.N., Grines V.Z., “Diffeomorfizmy s orientiruemymi geteroklinicheskimi mnozhestvami na dvumernykh mnogoobraziyakh”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. temat. sb. nauch. tr., Gork. gos. un-t, Gorkii, 1985, 139–152

[6] A. N. Bezdenezhnykh and V. Z. Grines, “Realization of gradient-like diffeomorphisms of two-dimensional manifolds”, Sel. Math. Sov., 11:1 (1992), 19–23 | MR | MR

[7] A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II”, Sel. Math. Sov., 11:1 (1992), 13–17 | MR | MR

[8] Casson A.J., Bleiler S.A., Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Stud. Texts, 9, Cambridge Univ. Press, Cambridge, 1988

[9] Constantin A., Kolev B., “The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere”, Enseign. math. Sér. 2, 40:3–4 (1994), 193–204; arXiv: math/0303256 [math.GN]

[10] V. Z. Grines and E. D. Kurenkov, “Diffeomorphisms of 2-manifolds with one-dimensional spaciously situated basic sets”, Izv. Math., 84:5 (2020), 862–909 | DOI | MR

[11] Grines V.Z., Medvedev T.V., Pochinka O.V., Dynamical systems on 2- and 3-manifolds, Dev. Math., 46, Springer, Cham, 2016

[12] Morozov A., Pochinka O., “Morse–Smale surfaced diffeomorphisms with orientable heteroclinic”, J. Dyn. Control Syst., 26:4 (2020), 629–639 | DOI | MR

[13] Nielsen J., Die Struktur periodischer Transformationen von Flächen, Danske Vidensk. Selsk. Math.-Fys. Medd., 15, Levin Munksgaard, København, 1937

[14] Smale S., “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 | DOI | MR

[15] A. Yu. Zhirov and R. V. Plykin, “On the relationship between one-dimensional hyperbolic attractors of surface diffeomorphisms and generalized pseudo-Anosov diffeomorphisms”, Math. Notes, 58:1 (1995), 779–781 | DOI | MR