Proceedings of the Steklov Institute of Mathematics

, Volume 270, Issue 1, pp 132–140 | Cite as

Gradient flows with wildly embedded closures of separatrices

  • E. V. ZhuzhomaEmail author
  • V. S. Medvedev


We show that for any n ≥ 4 there exists an n-dimensional closed manifold M n on which one can define a Morse-Smale gradient flow f t with two nodes and two saddles such that the closure of the separatrix of some saddle of f t is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.


Equilibrium Point STEKLOV Institute Invariant Manifold Stable Manifold Tubular Neighborhood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. V. Anosov, “Basic Concepts. Elementary Theory,” in Dynamical Systems-1 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 1, pp. 156–178, 178–204; Engl. transl. in Dynamical Systems I (Springer, Berlin, 1988), Encycl. Math. Sci. 1.Google Scholar
  2. 2.
    D. V. Anosov and V. V. Solodov, “Hyperbolic Sets,” in Dynamical Systems-9 (VINITI, Moscow, 1991), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 66, pp. 12–99; Engl. transl. in Dynamical Systems IX (Springer, Berlin, 1995), Encycl. Math. Sci. 66.Google Scholar
  3. 3.
    C. Bonatti, V. Z. Grines, V. S. Medvedev, and E. Pécou, “On the Topological Classification of Gradientlike Diffeomorphisms without Heteroclinic Curves on Three-Dimensional Manifolds,” Dokl. Akad. Nauk 377(2), 151–155 (2001) [Dokl. Math. 63 (2), 161–164 (2001)].MathSciNetGoogle Scholar
  4. 4.
    C. Bonatti, V. Z. Grines, V. S. Medvedev, and E. Pécou, “OnMorse-Smale Diffeomorphisms without Heteroclinic Intersections on Three-Manifolds,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 236, 66–78 (2002) [Proc. Steklov Inst. Math. 236, 58–69 (2002)].Google Scholar
  5. 5.
    C. Bonatti, V. Z. Grines, and O. V. Pochinka, “Classification of Morse-Smale Diffeomorphisms with a Finite Set of Heteroclinic Orbits on 3-Manifolds,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 250, 5–53 (2005) [Proc. Steklov Inst. Math. 250, 1–46 (2005)].MathSciNetGoogle Scholar
  6. 6.
    V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “OnMorse-Smale Diffeomorphisms with Four Periodic Points on Closed Orientable Manifolds,” Mat. Zametki 74(3), 369–386 (2003) [Math. Notes 74, 352–366 (2003)].MathSciNetGoogle Scholar
  7. 7.
    J. Palis, Jr. and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction (Springer, New York, 1982; Mir, Moscow, 1986).zbMATHGoogle Scholar
  8. 8.
    L. S. Pontryagin, Smooth Manifolds and Their Applications in Homotopy Theory (Akad. Nauk SSSR, Moscow, 1955), Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 45.Google Scholar
  9. 9.
    A. V. Chernavskii, “Singular Points of Topological Imbeddings of Manifolds and the Union of Locally Flat Cells,” Dokl. Akad. Nauk SSSR 167(3), 528–530 (1966) [Sov. Math., Dokl. 7, 433–436 (1966)].MathSciNetGoogle Scholar
  10. 10.
    J. J. Andrews and M. L. Curtis, “Knotted 2-Spheres in the 4-Sphere,” Ann. Math., Ser. 2, 70(3), 565–571 (1959).CrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Artin, “Zur Isotopie zweidimensionaler Flächen im R 4,” Abh. Math. Semin. Univ. Hamburg 4, 174–177 (1925).zbMATHCrossRefGoogle Scholar
  12. 12.
    C. Bonatti and V. Grines, “Knots as Topological Invariant for Gradient-like Diffeomorphisms of the Sphere S 3,” J. Dyn. Control Syst. 6, 579–602 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. Bonatti, V. Grines, V. Medvedev, and E. Pécou, “Three-Manifolds Admitting Morse-Smale Diffeomorphisms without Heteroclinic Curves,” Topology Appl. 117, 335–344 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    C. Bonatti, V. Grines, V. Medvedev, and E. Pécou, “Topological Classification of Gradient-like Diffeomorphisms on 3-Manifolds,” Topology 43, 369–391 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Eells, Jr. and N. H. Kuiper, “Manifolds Which Are Like Projective Planes,” Publ. Math., Inst. Hautes Étud. Sci. 14, 5–46 (1962).CrossRefMathSciNetGoogle Scholar
  16. 16.
    C. McA. Gordon and J. Luecke, “Knots Are Determined by Their Complements,” J. Am. Math. Soc. 2, 371–415 (1989).zbMATHMathSciNetGoogle Scholar
  17. 17.
    M. W. Hirsch, Differential Topology (Springer, New York, 1976), Grad. Texts Math. 33.zbMATHGoogle Scholar
  18. 18.
    P. B. Kronheimer and T. S. Mrowka, “Witten’s Conjecture and Property P,” Geom. Topol. 8, 295–310 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    D. Pixton, “Wild Unstable Manifolds,” Topology 16, 167–172 (1977).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    G. Reeb, “Sur certaines propriétés topologiques des variétés feuilletées,” in W.-T. Wu and G. Reeb, Sur les espaces fibrés et les variétés feuilletées (Hermann, Paris, 1952), Actual. Sci. Ind. 1183, pp. 91–154.Google Scholar
  21. 21.
    C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd ed. (CRC Press, Boca Raton, FL, 1999), Stud. Adv. Math.zbMATHGoogle Scholar
  22. 22.
    S. Smale, “On Gradient Dynamical Systems,” Ann. Math., Ser. 2, 74, 199–206 (1961).CrossRefMathSciNetGoogle Scholar
  23. 23.
    S. Smale, “Differentiable Dynamical Systems,” Bull. Am. Math. Soc. 73(6), 747–817 (1967) [Usp. Mat. Nauk 25 (1), 113–185 (1970)].CrossRefMathSciNetGoogle Scholar
  24. 24.
    J. Stallings, “On Topologically Unknotted Spheres,” Ann. Math., Ser. 2, 77, 490–503 (1963).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Nizhni Novgorod State Pedagogical UniversityNizhni NovgorodRussia
  2. 2.Research Institute for Applied Mathematics and CyberneticsLobachevsky State University of Nizhni NovgorodNizhni NovgorodRussia

Personalised recommendations