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

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Abstract

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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References

  1. W. Abikoff, “The uniformization theorem,” Am. Math. Mon. 88 (8), 574–592 (1981).

    Article  MathSciNet  Google Scholar 

  2. S. Kh. Aranson and V. Z. Grines, “The topological classification of cascades on closed two-dimensional manifolds,” Russ. Math. Surv. 45 (1), 1–35 (1990) [transl. from Usp. Mat. Nauk 45 (1), 3–32 (1990)].

    Article  MathSciNet  Google Scholar 

  3. A. N. Bezdenezhnykh, “Topological classification of Morse–Smale diffeomorphisms with an orientable heteroclinic set on two-dimensional manifolds,” Cand. Sci. (Phys.–Math.) Dissertation (Gork. Gos. Univ., Gorky, 1985).

    Google Scholar 

  4. A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I,” in Methods of the Qualitative Theory of Differential Equations (Gork. Gos. Univ., Gorky, 1984), pp. 22–38. Engl. transl.: Sel. Math. Sov. 11 (1), 1–11 (1992).

    Google Scholar 

  5. A. N. Bezdenezhnykh and V. Z. Grines, “Diffeomorphisms with orientable heteroclinic sets on two-dimensional manifolds,” in Methods of the Qualitative Theory of Differential Equations (Gork. Gos. Univ., Gorky, 1985), pp. 139–152 [in Russian].

    Google Scholar 

  6. A. N. Bezdenezhnykh and V. Z. Grines, “Realization of gradient-like diffeomorphisms of two-dimensional manifolds,” in Differential and Integral Equations (Gork. Gos. Univ., Gorky, 1985), pp. 33–37. Engl. transl.: Sel. Math. Sov. 11 (1), 19–23 (1992).

    Google Scholar 

  7. A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II,” in Methods of the Qualitative Theory of Differential Equations (Gork. Gos. Univ., Gorky, 1987), pp. 24–31. Engl. transl.: Sel. Math. Sov. 11 (1), 13–17 (1992).

    Google Scholar 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  10. V. Z. Grines and E. D. Kurenkov, “Diffeomorphisms of 2-manifolds with one-dimensional spaciously situated basic sets,” Izv. Math. 84 (5), 862–909 (2020) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 84 (5), 40–97 (2020)].

    Article  MathSciNet  Google Scholar 

  11. V. Z. Grines, T. V. Medvedev, and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds (Springer, Cham, 2016), Dev. Math. 46.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  14. S. Smale, “Differentiable dynamical systems,” Bull. Am. Math. Soc. 73 (6), 747–817 (1967).

    Article  MathSciNet  Google Scholar 

  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), 779–781 (1995) [transl. from Mat. Zametki 58 (1), 149–152 (1995)].

    Article  MathSciNet  Google Scholar 

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Funding

This work, except for Section 1.2, was supported by the Russian Science Foundation under grant 17-11-01041. The work presented in Section 1.2 was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (contract no. 19-7-1-15-1).

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Authors and Affiliations

  1. International Laboratory of Dynamical Systems and Applications, National Research University Higher School of Economics, Bol’shaya Pecherskaya ul. 25/12, Nizhny Novgorod, 603155, Russia

    V. Z. Grines, A. I. Morozov & O. V. Pochinka

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  1. V. Z. Grines
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  2. A. I. Morozov
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  3. O. V. Pochinka
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Correspondence to V. Z. Grines.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 95–107 https://doi.org/10.4213/tm4234.

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Grines, V.Z., Morozov, A.I. & Pochinka, O.V. Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection. Proc. Steklov Inst. Math. 315, 85–97 (2021). https://doi.org/10.1134/S0081543821050072

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