Determination of the Homotopy Type of a Morse-Smale Diffeomorphism on an Orientable Surface by a Heteroclinic Intersection

• Autores: Vyacheslav Grines, Andrei Morozov , Olga Pochinka
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 3, 2023
• Idioma: inglés
• DOI: 10.1007/s12346-023-00809-9
• Enlaces
  • Texto completo
• Resumen

  • This paper is devoted to the study of homotopy types of orientation-preserving Morse-Smale diffeomorphisms on closed orientable surfaces. Since any Morse-Smale diffeomorphism has a finite set of periodic points, then, according to the Nielsen–Thurston classification, it is homotopic to either a periodic homeomorphism or an algebraically finite order homeomorphism. It follows from the results of V. Grines and A. Bezdenezhnykh that any gradient-like diffeomorphism is homotopic to a periodic homeomorphism. However, when the wandering set of a given diffeomorphism contains heteroclinic intersections, then the question of its homotopy type is remains open. In the present work, an algorithm for recognizing the homotopy type of a non-gradient-like Morse-Smale diffeomorphism by its heteroclinic intersection is proposed. The algorithm is based on the construction of a filtration for a diffeomorphism and calculation of the intersection index of saddle separatrices in the fundamental annuli of filtration elements. It is established that a Morse-Smale diffeomorphism is homotopic to a periodic homeomorphism if and only if the total intersection index over all homotopic annuli is equal to zero.

• Referencias bibliográficas
  • 1. Anosov, D.V., Aranson, S. Kh., Grines, V.Z., Plykin, R.V., Sataev, E.A., Safonov, A.V., Solodov, V.V., Starkov, A.N., Stepin, A.M., Shlyachkov,...
  • 2. Banyaga, A.: On the structure of the group of equivariant diffeomorphisms. Topology 16(3), 279–283 (1977). https://doi.org/10.1016/0040-9383(77)90009-X
  • 3. Bezdenezhykh, A.N., Grines, V.Z.: Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional...
  • 4. Bezdenezhykh, A.N., Grines, V.Z.: Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional...
  • 5. Bezdenezhykh, A.N., Grines, V.Z.: Realization of gradient-like diffeomorphisms of two-dimensional manifolds. Sel. Math. Sov. 11(1), 19–23...
  • 6. Casson, A.J., Bleiler, S.A.: Automorphisms of surfaces after Nielsen and Thurston. London Math. Soc. Student Texts, Cambridge University...
  • 7. Farb, B., Margalit, D.: A primer on mapping class groups (pms-49). Princeton university press, Princeton (2011)
  • 8. Grines, V.Z., Kurenkov. E.D.: Diffeomorphisms of two-dimensional manifolds with one-dimensional spaciously located basic sets. Izv. RAN....
  • 9. Grines, V.Z., Medvedev, T.V., Pochinka, O.V.: Dynamical systems on 2-and 3-manifolds. Springer, Cham, 46 (2016)
  • 10. Grines, V.Z., Morozov, A.I., Pochinka, O.V.: Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse-Smale Diffeomorphisms...
  • 11. Grines, V.Z., Zhuzhoma, E.V., Medvedev, V.S., Pochinka, O.V.: Global attractor and repeller of MorseSmale diffeomorphisms. Proc Steklov...
  • 12. Malyshev D., Morozov A., Pochinka, O.: Combinatorial invariant for Morse-Smale diffeomorphisms on surfaces with orientable heteroclinic....
  • 13. Morozov, A.I.: Determination of the homotopy type of a Morse—smale diffeomorphism on a 2-torus by heteroclinic intersection. Russian J....
  • 14. Nielsen, J.: Die Structur periodisher Transformationen von Flachen. D.K. Dan. Vidensk. Selsk. Math fys. Medd. 15, 1–77 (1937)
  • 15. Nielsen, J.: Surface transformation classes of algebraically finite type. Danske Vid. Selsk. Mat. Fys. Medd. 21, 89 (1944)
  • 16. Nielsen, J.: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. II Acta Math. 53, 1–76 (1929)
  • 17. Nielsen, J.: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. III Acta Math. 58, 87–167 (1932)
  • 18. Nielsen, J. Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. I Acta Math. 50, 189–358 (1927)
  • 19. Pixton, D.: Wild unstable manifolds. Topology 16(2), 167–172 (1977). https://doi.org/10.1007/978-3-319-44847-3-4
  • 20. Rolfsen, D.: Knots and links. Mathematics Lecture Series, 7 (1990)
  • 21. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967). https://doi.org/10.1090/S0002-9904-1967-11798-1
  • 22. Zhirov, AYu., Plykin, R.V.: Correspondence between one-dimensional hyperbolic attractors of diffeomorphisms of surfaces and generalized...