Skip to main content
SpringerLink
Log in

Morse–Smale Systems and Topological Structure of Carrier Manifolds

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We review the results describing the connection between the global dynamics of Morse–Smale systems on closed manifolds and the topology of carrier manifolds. Also we consider the results related to topological classification of Morse–Smale systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. S. Afraymovich and L.P. Shil’nikov, “On singular sets of Morse–Smale systems,” Tr. Mosk. Mat. Obs., 28, 181–214 (1973).

    MathSciNet  Google Scholar 

  2. A. A. Andronov, E.A. Leontovich, I. I. Gordon, and A. G. Mayer, Qualitative Theory of Second-Order Dynamical Systems [in Russian], Nauka, Moscow (1966).

  3. A. A. Andronov and L. S. Pontryagin, “Rough systems,” C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 14, No. 5, 247–250 (1937).

    Google Scholar 

  4. D. V. Anosov, “Roughness of geodesic flows on compact Riemann manifolds of negative curvature,” Dokl. Akad. Nauk SSSR, 145, No. 4, 707–709 (1962).

    MathSciNet  Google Scholar 

  5. D. V. Anosov, “Geodesic flows on close Riemann manifolds of negative curvature,” Trudy Mat. Inst. Steklov, 90 (1967).

  6. S. H. Aranson, “Trajectories on nonoriented two-dimensional manifolds,” Mat. Sb. (N.S.), 80 (122), No. 3, 314–333 (1969).

    Google Scholar 

  7. S. Aranson, G. Belitsky, and E. Zhuzhoma, Introduction to Qualitative Theory of Dynamical Systems on Closed Surfaces, AMS, Providence (1996).

  8. S.H. Aranson and V. S. Medvedev, “Regular components of homeomorphisms of n-dimensional sphere,” Mat. Sb. (N.S.), 85 (127), 3–17 (1971).

    MathSciNet  Google Scholar 

  9. E. Artin and R. H. Fox, “Some wild cells and spheres in three-dimensional space,” Ann. of Math. (2), 49, 979–990 (1948).

    MathSciNet  MATH  Google Scholar 

  10. D. Asimov, “Round handles and non-singular Morse–Smale flows,” Ann. of Math. (2), 102, 41–54 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Asimov, “Homotopy of non-singular vector fields to structurally stable ones,” Ann. of Math. (2), 102, 55–65 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Batterson, “The dynamics of Morse–Smale diffeomorphisms on the torus,” Trans. Amer. Math. Soc., 256, 395–403 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Batterson, “Orientation reversing Morse–Smale diffeomorphisms on the torus,” Trans. Amer. Math. Soc., 264, 29–37 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Batterson, M. Handel, and C. Narasimhan, “Orientation reversing Morse–Smale diffeomorphisms of S 2,” Invent. Math., 64, 345–356 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. N. Bezdenezhnykh and V. Z. Grines, “Realization of gradient-like diffeomorphisms of twodimensional manifolds,” Differential and Integral Equations, Gor’kiy State Univ., Gor’kiy, 33–37 (1985).

  16. A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I,” KTDU Methods, Gor’kiy, 22–38 (1985).

  17. A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II,” KTDU Methods, Gor’kiy, 24–32 (1987).

  18. F. Béguin, “Smale diffeomorphisms of surfaces: an algorithm for the conjugacy problem,” preprint 1999).

  19. Yu. Bin, “Behavior 0 nonsingular Morse—Smale flows on S 3,” Discrete Contin. Dyn. Syst., 36, No. 1, 509–540 (2016).

    MathSciNet  MATH  Google Scholar 

  20. P. Blanchard and J. Franks, “The dynamical complexity of orientation reversing homeomorphisms of surfaces,” Invent. Math., 62, 333–339 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  21. Ch. Bonatti and V. Grines, “Knots as topological invariant for gradient-like diffeomorphisms of the sphere S 3,” J. Dyn. Control Syst., 6, No. 4, 579–602 (2000).

    Article  MATH  Google Scholar 

  22. Ch. Bonatti, V. Z. Grines, V. S. Medvedev, and E. Peku, “On topological classification of gradientlike diffeomorphisms without heteroclinic curves on three-dimensional manifolds,” Dokl. Akad. Nauk, 377, No. 2, 151–155 (2001).

    MathSciNet  MATH  Google Scholar 

  23. Ch. Bonatti, V. Z. Grines, V. S. Medvedev, and E. Peku, “On Morse–Smale diffeomorphisms without heteroclinic intersections on three-dimensional manifolds,” Tr. Mat. Inst. Steklova, 236, 66–78 (2002).

    MathSciNet  MATH  Google Scholar 

  24. Ch. Bonatti, V. Grines, V. Medvedev, and E. Pecou, “Three-dimensional manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves,” Topology Appl., 117, 335–344 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  25. Ch. Bonatti, V. Grines, V. Medvedev, and E. Pecou, “Topological classification of gradient-like diffeomorphisms on 3-manifolds,” Topology, 43, 369–391 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  26. Ch. Bonatti, V. Z. Grines, and O.V. Pochinka, “Classification of Morse–Smale diffeomorphisms with finite set of heteroclinic orbits on 3-manifolds,” Tr. Mat. Inst. Steklova, 250, 5–53 (2005).

    MathSciNet  MATH  Google Scholar 

  27. Ch. Bonatti, V. Grines, and O. Pochinka, “Classification of Morse–Smale diffeomorphisms with the chain of saddles on 3-manifolds,” Foliations 2005, World Scientific, Singapore, 121–147 (2006).

  28. Ch. Bonatti and R. Langevin, “Difféomorphismes de Smale des surfaces,” Astérisque, No. 250 (1998).

  29. R. Bowen, “Periodic points and measures for axiom A diffeomorphisms,” Trans. Amer. Math. Soc., 154, 337–397 (1971).

  30. J.C. Cantrell and C.H. Edwards, “Almost locally polyhedral curves in Euclidean n-space,” Trans. Amer. Math. Soc., 107, 451–457 (1963).

    MathSciNet  MATH  Google Scholar 

  31. A. Cobham, “The intrinsic computational difficulty of functions,” International Congress for Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam, 24–30 (1964).

  32. H. Debrunner and R. Fox, “A mildly wild imbedding of an n-frame,” Duke Math. J., 27, 425–429 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Fleitas, “Classification of gradient-like flows in dimension two and three,” Bol. Soc. Brasil. Mat., 2, No. 6, 155–183 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  34. A. T. Fomenko and D. B. Fuks, Course in Homotopical Topology [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  35. J. Franks, “Some maps with infinitely many hyperbolic periodic points,” Trans. Amer. Math. Soc., 226, 175–179 (1977).

    MathSciNet  MATH  Google Scholar 

  36. J. Franks, “The periodic structure of non-singular Morse–Smale flows,” Comment. Math. Helv., 53, 279–294 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  37. J. M. Franks, Homology and Dynamical Systems, AMS, Providence (1982).

  38. V. Z. Grines, “Topological classification of Morse–Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces,” Mat. Zametki, 54, 3–17 (1993).

    MathSciNet  MATH  Google Scholar 

  39. V. Z. Grines, E. Ya. Gurevich, and V. S. Medvedev, “On classification of Morse–Smale diffeomorphisms with one-dimensional set of nonstable separatrices,” Tr. Mat. Inst. Steklova, 270, 20–35 (2010).

    MATH  Google Scholar 

  40. V. Grines, E. Gurevich, and O. Pochinka, “Topological classification of Morse–Smale diffeomorphisms without heteroclinic intersection,” J. Math. Sci. (N.Y.), 208, No. 1, 81–91 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  41. V. Z. Grines, S. Kh. Kapkaeva, and O. V. Pochinka, “Three-colored graph as complete topological invariant for gradient-like diffeomorphisms of surfaces,” Mat. Sb., 205, No. 10, 19–46 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  42. V. Grines, D. Malyshev, O. Pochinka, and S. Zinina, “Efficient algorithms for the recognition of topologically conjugate gradient-like diffeomorphisms,” Regul. Chaotic Dyn., 21, No 2, 189–203 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  43. V. Grines, T. Medvedev, O. Pochinka, and E. Zhuzhoma, “On heteroclinic separators of magnetic fields in electrically conducting fluids,” Phys. D., 294, 1–5 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  44. V. Z. Grines and O.V. Pochinka, Introduction to Topological Classification of Diffeomorphisms on Manifolds of Dimensions Two and Three [in Russian], NITS “Regulyarnaya i Khaoticheskaya Dinamika”, Moscow–Izhevsk (2011).

  45. V. Z. Grines, E.V. Zhuzhoma, and V. S. Medvedev, “New relations for Morse–Smale flows and diffeomorphisms,” Dokl. Akad. Nauk, 382, No. 6, 730–733 (2002).

    MathSciNet  MATH  Google Scholar 

  46. V. Z. Grines, E.V. Zhuzhoma, and V. S. Medvedev, “New relations for Morse–Smale systems with trivially embedded one-dimensional separatrices,” Mat. Sb., 194, No. 7, 25–56 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  47. V. Z. Grines, E.V. Zhuzhoma, and V. S. Medvedev, “On Morse–Smale diffeomorphisms with four periodic points on closed oriented manifolds,” Mat. Zametki, 74, No. 3, 369–386 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  48. V. Z. Grines, E.V. Zhuzhoma, V. S. Medvedev, and O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms,” Tr. Mat. Inst. Steklova, 271, 111–133 (2010).

    MathSciNet  MATH  Google Scholar 

  49. D. M. Grobman, “On the diffeomorphism of systems of differential equations,” Dokl. Akad. Nauk, 128, No. 5, 880–881 (1959).

    MathSciNet  MATH  Google Scholar 

  50. D. M. Grobman, “Topological classification of neighborhoods of a singular point in n-dimensional space,” Mat. Sb., 56, No. 1, 77–94 (1962).

    MathSciNet  Google Scholar 

  51. E.Ya. Gurevich, “On Morse–Smale diffeomorphisms on manifolds of dimension greater than three,” Tr. Srednevolzhsk. Mat. Obs., 5, No. 1, 161–165 (2003).

    MathSciNet  Google Scholar 

  52. E.Ya. Gurevich and V. S. Medvedev, “On n-dimensional manifolds allowing diffeomorphisms with saddle points of indices 1 and n − 1,” Tr. Srednevolzhsk. Mat. Obs., 8, No. 1, 204–208 (2006).

    MATH  Google Scholar 

  53. C. Gutierrez, “Structural stability for flows on the torus with a cross-cap,” Trans. Amer. Math. Soc., 241, 311–320 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Handel, “The entropy of orientation reversing homeomorphisms of surfaces,” Topology, 21, 291–296 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  55. O. G. Harrold, H.C. Griffith, and E.E. Posey, “A characterization of tame curves in three-space,” Trans. Amer. Math. Soc., 79, 12–34 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  56. P. Hartman, “On the local linearization of differential equations,” Proc. Amer. Math. Soc., 14, No. 4, 568–573 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  57. M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Springer, Berlin–Heidelberg–New York (1977).

  58. E. A. Leontovich and A.G. Mayer, “On trajectories determining qualitative structure of partition of a sphere into trajectories,” Dokl. AN SSSR, 14, No. 5, 251–257 (1937).

    Google Scholar 

  59. E. A. Leontovich and A.G. Mayer, “On the scheme determining topological structure of partition into trajectories,” Dokl. AN SSSR, 103, No. 4, 557–560 (1955).

    Google Scholar 

  60. N. G. Markley, “The Poincare–Bendixon theorem for the Klein bottle,” Trans. Amer. Math. Soc., 135, 159–165 (1969).

    MathSciNet  MATH  Google Scholar 

  61. S. V. Matveev, “Classification of sufficiently large three-dimensional manifolds,” Uspekhi Mat. Nauk, 52, No. 5, 147–174 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  62. A. G. Mayer, “Rough transformation of a circle to a circle,” Sci. Notes Gor’kiy State Univ., 12, 215–229 (1939).

    Google Scholar 

  63. V. Medvedev and E. Zhuzhoma, “Morse–Smale systems with few non-wandering points,” Topology Appl., 160, No. 3, 498–507 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  64. V. S. Medvedev and E.V. Zhuzhoma, “Continuous Morse–Smale flows with three equilibrium states,” to appear in Mat. Sb.

  65. T. M. Mitryakova and O.V. Pochinka, “On necessary and sufficient conditions of topological conjugacy of diffeomorphisms of surfaces with finite number of orbits of heteroclinic tangency,” Tr. Mat. Inst. Steklova, 270, 198–219 (2010).

    MATH  Google Scholar 

  66. J. W. Morgan, “Non-singular Morse–Smale flows on 3-dimensional manifolds,” Topology, 18, 41–53 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  67. M. Morse, Calculus of Variations in the Large, Interscience Publ., New York (1934).

    Book  MATH  Google Scholar 

  68. C. Narasimhan, “The periodic behavior of Morse–Smale diffeomorphisms on compact surfaces,” Trans. Amer. Math. Soc., 248, 145–169 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  69. I. Nikolaev, “Graphs and flows on surfaces,” Ergodic Theory Dynam. Systems, 18, 207–220 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  70. I. Nikolaev and E. Zhuzhoma, Flows on 2-Dimensional Manifolds, Springer, Berlin (1999).

    Book  MATH  Google Scholar 

  71. A. A. Oshemkov and V.V. Sharko, “On classification of Morse–Smale flows on two-dimensional manifolds,” Mat. Sb., 189, No. 8, 93–140 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  72. J. Palis, “On Morse–Smale dynamical systems,” Topology, 8, No. 4, 385–404 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  73. J. Palis and S. Smale, “Structural stability theorems,” Global Analysis, AMS, Providence, 223–231 (1970).

  74. M. M. Peixoto, “On structural stability,” Ann. of Math. (2), 69, 199–222 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  75. M. M. Peixoto, “Structural stability on two-dimensional manifolds,” Topology, 1, 101–120 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  76. M. M. Peixoto, “Structural stability on two-dimensional manifolds. A further remark,” Topology, 2, 179-180 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  77. M. M. Peixoto, “On a classification of flows on 2-manifolds,” Dynamical Systems, Academic Press, New York, 389–492 (1973)

  78. D. Pixton, “Wild unstable manifolds,” Topology, 16, 167–172 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  79. V. A. Pliss, “On roughness of differential equations set on a torus,” Vestnik Leningrad. Univ., 13, 15–23 (1960).

    Google Scholar 

  80. O. V. Pochinka, “Classification of Morse–Smale diffeomorphisms on 3-manifolds,” Dokl. Akad. Nauk, 440, No. 6, 34–37 (2011).

    MATH  Google Scholar 

  81. A. Prishlyak, “Morse–Smale vector fields without closed trajectories on three-dimensional manifolds,” Mat. Zametki, 71, No. 2, 230–235 (2002).

    MathSciNet  MATH  Google Scholar 

  82. K. Sasano, “Links of closed orbits of non-singular Morse–Smale flows,” Proc. Amer. Math. Soc., 88, 727–734 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  83. M. Shub, “Morse–Smale diffeomorphisms are unipotent on homology,” Dynamical Systems, Academic Press, New York, 489–491 (1973).

  84. M. Shub and D. Sullivan, “Homology theory and dynamical systems,” Topology, 4, 109–132 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  85. Ya.G. Sinay, “Markov partitions and U-diffeomorphisms,” Funktsional. Anal. i Prilozhen., 2, No. 1, 64–89 (1968).

    MathSciNet  Google Scholar 

  86. Ya.G. Sinay, “Construction of Markov partitions,” Funkts. analiz i ego prilozh., 2, No. 3, 70–80 (1968).

    Google Scholar 

  87. S. Smale, “Morse inequalities for a dynamical system,” Bull. Amer. Math. Soc., 66, 43–49 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  88. S. Smale, “Generalized Poincare’s conjecture in dimensions greater than four,” Bull. Amer. Math. Soc., 66, 485–488 (1960).

    Article  MathSciNet  Google Scholar 

  89. S. Smale, “On gradient dynamical systems,” Ann. of Math. (2), 74, 199–206 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  90. S. Smale, “Generalized Poincare’s conjecture in dimensions greater than four,” Ann. of Math. (2), 74, 391–406 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  91. S. Smale, “Diffeomorphisms with many periodic points,” Differ. and Combinat. Topology, Sympos., Marston Morse, Princeton, 63–80 (1965).

  92. S. Smale, “Differentiable dynamical systems,” Uspekhi Mat. Nauk, 25, No. 1, 113–185 (1970).

    MathSciNet  MATH  Google Scholar 

  93. Ya. L. Umanskiy, “Necessary and sufficient conditions for topological equivalence of threedimensional Morse–Smale dynamical systems with finite number of singular trajectories,” Mat. Sb., 181, No. 2, 212–239 (1990).

    Google Scholar 

  94. M. Wada, “Closed orbits of non-singular Morse–Smale flows on S 3,” J. Math. Soc. Japan, 41, 405–413 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  95. X. Wang, “The C *-algebras of Morse–Smale flows on two-manifolds,” Ergodic Theory Dynam. Systems, 10, 565–597 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  96. K. Yano, “A note on non-singular Morse–Smale flows on S 3,” Proc. Japan Acad. Ser. A Math. Sci., 58, 447–450 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  97. E.V. Zhuzhoma and V. S. Medvedev, “Morse–Smale systems with three nonwandering points,” Dokl. Akad. Nauk, 440, No. 1, 11–14 (2011).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research University “Higher School of Economics” in Nizhniy Novgorod, Nizhniy Novgorod, Russia

    V. Z. Grines, Ye. V. Zhuzhoma & O. V. Pochinka

  2. Lobachevski State University of Nizhniy Novgorod, Nizhniy Novgorod, Russia

    V. Z. Grines

Authors
  1. V. Z. Grines
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ye. V. Zhuzhoma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. V. Pochinka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Z. Grines.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 61, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grines, V.Z., Zhuzhoma, Y.V. & Pochinka, O.V. Morse–Smale Systems and Topological Structure of Carrier Manifolds. J Math Sci 239, 549–581 (2019). https://doi.org/10.1007/s10958-019-04316-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-019-04316-8

Search

Navigation