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Journal of Mathematical Sciences

, Volume 239, Issue 5, pp 549–581 | Cite as

Morse–Smale Systems and Topological Structure of Carrier Manifolds

  • V. Z. GrinesEmail author
  • Ye. V. Zhuzhoma
  • O. V. Pochinka
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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.

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References

  1. 1.
    V. S. Afraymovich and L.P. Shil’nikov, “On singular sets of Morse–Smale systems,” Tr. Mosk. Mat. Obs., 28, 181–214 (1973).MathSciNetGoogle Scholar
  2. 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).Google Scholar
  3. 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. 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).MathSciNetGoogle Scholar
  5. 5.
    D. V. Anosov, “Geodesic flows on close Riemann manifolds of negative curvature,” Trudy Mat. Inst. Steklov, 90 (1967).Google Scholar
  6. 6.
    S. H. Aranson, “Trajectories on nonoriented two-dimensional manifolds,” Mat. Sb. (N.S.), 80 (122), No. 3, 314–333 (1969).Google Scholar
  7. 7.
    S. Aranson, G. Belitsky, and E. Zhuzhoma, Introduction to Qualitative Theory of Dynamical Systems on Closed Surfaces, AMS, Providence (1996).Google Scholar
  8. 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).MathSciNetGoogle Scholar
  9. 9.
    E. Artin and R. H. Fox, “Some wild cells and spheres in three-dimensional space,” Ann. of Math. (2), 49, 979–990 (1948).MathSciNetzbMATHGoogle Scholar
  10. 10.
    D. Asimov, “Round handles and non-singular Morse–Smale flows,” Ann. of Math. (2), 102, 41–54 (1975).MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Asimov, “Homotopy of non-singular vector fields to structurally stable ones,” Ann. of Math. (2), 102, 55–65 (1975).MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. Batterson, “The dynamics of Morse–Smale diffeomorphisms on the torus,” Trans. Amer. Math. Soc., 256, 395–403 (1979).MathSciNetzbMATHGoogle Scholar
  13. 13.
    S. Batterson, “Orientation reversing Morse–Smale diffeomorphisms on the torus,” Trans. Amer. Math. Soc., 264, 29–37 (1981).MathSciNetzbMATHGoogle Scholar
  14. 14.
    S. Batterson, M. Handel, and C. Narasimhan, “Orientation reversing Morse–Smale diffeomorphisms of S 2,” Invent. Math., 64, 345–356 (1981).MathSciNetzbMATHGoogle Scholar
  15. 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).Google Scholar
  16. 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).Google Scholar
  17. 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).Google Scholar
  18. 18.
    F. Béguin, “Smale diffeomorphisms of surfaces: an algorithm for the conjugacy problem,” preprint 1999).Google Scholar
  19. 19.
    Yu. Bin, “Behavior 0 nonsingular Morse—Smale flows on S 3,” Discrete Contin. Dyn. Syst., 36, No. 1, 509–540 (2016).MathSciNetzbMATHGoogle Scholar
  20. 20.
    P. Blanchard and J. Franks, “The dynamical complexity of orientation reversing homeomorphisms of surfaces,” Invent. Math., 62, 333–339 (1980).MathSciNetzbMATHGoogle Scholar
  21. 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).zbMATHGoogle Scholar
  22. 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).MathSciNetzbMATHGoogle Scholar
  23. 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).MathSciNetzbMATHGoogle Scholar
  24. 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).MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ch. Bonatti, V. Grines, V. Medvedev, and E. Pecou, “Topological classification of gradient-like diffeomorphisms on 3-manifolds,” Topology, 43, 369–391 (2004).MathSciNetzbMATHGoogle Scholar
  26. 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).MathSciNetzbMATHGoogle Scholar
  27. 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).Google Scholar
  28. 28.
    Ch. Bonatti and R. Langevin, “Difféomorphismes de Smale des surfaces,” Astérisque, No. 250 (1998).Google Scholar
  29. 29.
    R. Bowen, “Periodic points and measures for axiom A diffeomorphisms,” Trans. Amer. Math. Soc., 154, 337–397 (1971).Google Scholar
  30. 30.
    J.C. Cantrell and C.H. Edwards, “Almost locally polyhedral curves in Euclidean n-space,” Trans. Amer. Math. Soc., 107, 451–457 (1963).MathSciNetzbMATHGoogle Scholar
  31. 31.
    A. Cobham, “The intrinsic computational difficulty of functions,” International Congress for Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam, 24–30 (1964).Google Scholar
  32. 32.
    H. Debrunner and R. Fox, “A mildly wild imbedding of an n-frame,” Duke Math. J., 27, 425–429 (1960).MathSciNetzbMATHGoogle Scholar
  33. 33.
    G. Fleitas, “Classification of gradient-like flows in dimension two and three,” Bol. Soc. Brasil. Mat., 2, No. 6, 155–183 (1975).MathSciNetzbMATHGoogle Scholar
  34. 34.
    A. T. Fomenko and D. B. Fuks, Course in Homotopical Topology [in Russian], Nauka, Moscow (1989).Google Scholar
  35. 35.
    J. Franks, “Some maps with infinitely many hyperbolic periodic points,” Trans. Amer. Math. Soc., 226, 175–179 (1977).MathSciNetzbMATHGoogle Scholar
  36. 36.
    J. Franks, “The periodic structure of non-singular Morse–Smale flows,” Comment. Math. Helv., 53, 279–294 (1978).MathSciNetzbMATHGoogle Scholar
  37. 37.
    J. M. Franks, Homology and Dynamical Systems, AMS, Providence (1982).Google Scholar
  38. 38.
    V. Z. Grines, “Topological classification of Morse–Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces,” Mat. Zametki, 54, 3–17 (1993).MathSciNetzbMATHGoogle Scholar
  39. 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).zbMATHGoogle Scholar
  40. 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).MathSciNetzbMATHGoogle Scholar
  41. 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).MathSciNetzbMATHGoogle Scholar
  42. 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).MathSciNetzbMATHGoogle Scholar
  43. 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).MathSciNetzbMATHGoogle Scholar
  44. 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).Google Scholar
  45. 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).MathSciNetzbMATHGoogle Scholar
  46. 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).MathSciNetzbMATHGoogle Scholar
  47. 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).MathSciNetzbMATHGoogle Scholar
  48. 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).MathSciNetzbMATHGoogle Scholar
  49. 49.
    D. M. Grobman, “On the diffeomorphism of systems of differential equations,” Dokl. Akad. Nauk, 128, No. 5, 880–881 (1959).MathSciNetzbMATHGoogle Scholar
  50. 50.
    D. M. Grobman, “Topological classification of neighborhoods of a singular point in n-dimensional space,” Mat. Sb., 56, No. 1, 77–94 (1962).MathSciNetGoogle Scholar
  51. 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).Google Scholar
  52. 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).zbMATHGoogle Scholar
  53. 53.
    C. Gutierrez, “Structural stability for flows on the torus with a cross-cap,” Trans. Amer. Math. Soc., 241, 311–320 (1978).MathSciNetzbMATHGoogle Scholar
  54. 54.
    M. Handel, “The entropy of orientation reversing homeomorphisms of surfaces,” Topology, 21, 291–296 (1982).MathSciNetzbMATHGoogle Scholar
  55. 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).MathSciNetzbMATHGoogle Scholar
  56. 56.
    P. Hartman, “On the local linearization of differential equations,” Proc. Amer. Math. Soc., 14, No. 4, 568–573 (1963).MathSciNetzbMATHGoogle Scholar
  57. 57.
    M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Springer, Berlin–Heidelberg–New York (1977).Google Scholar
  58. 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. 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. 60.
    N. G. Markley, “The Poincare–Bendixon theorem for the Klein bottle,” Trans. Amer. Math. Soc., 135, 159–165 (1969).MathSciNetzbMATHGoogle Scholar
  61. 61.
    S. V. Matveev, “Classification of sufficiently large three-dimensional manifolds,” Uspekhi Mat. Nauk, 52, No. 5, 147–174 (1997).MathSciNetzbMATHGoogle Scholar
  62. 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. 63.
    V. Medvedev and E. Zhuzhoma, “Morse–Smale systems with few non-wandering points,” Topology Appl., 160, No. 3, 498–507 (2013).MathSciNetzbMATHGoogle Scholar
  64. 64.
    V. S. Medvedev and E.V. Zhuzhoma, “Continuous Morse–Smale flows with three equilibrium states,” to appear in Mat. Sb. Google Scholar
  65. 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).zbMATHGoogle Scholar
  66. 66.
    J. W. Morgan, “Non-singular Morse–Smale flows on 3-dimensional manifolds,” Topology, 18, 41–53 (1979).MathSciNetzbMATHGoogle Scholar
  67. 67.
    M. Morse, Calculus of Variations in the Large, Interscience Publ., New York (1934).zbMATHGoogle Scholar
  68. 68.
    C. Narasimhan, “The periodic behavior of Morse–Smale diffeomorphisms on compact surfaces,” Trans. Amer. Math. Soc., 248, 145–169 (1979).MathSciNetzbMATHGoogle Scholar
  69. 69.
    I. Nikolaev, “Graphs and flows on surfaces,” Ergodic Theory Dynam. Systems, 18, 207–220 (1998).MathSciNetzbMATHGoogle Scholar
  70. 70.
    I. Nikolaev and E. Zhuzhoma, Flows on 2-Dimensional Manifolds, Springer, Berlin (1999).zbMATHGoogle Scholar
  71. 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).MathSciNetzbMATHGoogle Scholar
  72. 72.
    J. Palis, “On Morse–Smale dynamical systems,” Topology, 8, No. 4, 385–404 (1969).MathSciNetzbMATHGoogle Scholar
  73. 73.
    J. Palis and S. Smale, “Structural stability theorems,” Global Analysis, AMS, Providence, 223–231 (1970).Google Scholar
  74. 74.
    M. M. Peixoto, “On structural stability,” Ann. of Math. (2), 69, 199–222 (1959).MathSciNetzbMATHGoogle Scholar
  75. 75.
    M. M. Peixoto, “Structural stability on two-dimensional manifolds,” Topology, 1, 101–120 (1962).MathSciNetzbMATHGoogle Scholar
  76. 76.
    M. M. Peixoto, “Structural stability on two-dimensional manifolds. A further remark,” Topology, 2, 179-180 (1963).MathSciNetzbMATHGoogle Scholar
  77. 77.
    M. M. Peixoto, “On a classification of flows on 2-manifolds,” Dynamical Systems, Academic Press, New York, 389–492 (1973)Google Scholar
  78. 78.
    D. Pixton, “Wild unstable manifolds,” Topology, 16, 167–172 (1977).MathSciNetzbMATHGoogle Scholar
  79. 79.
    V. A. Pliss, “On roughness of differential equations set on a torus,” Vestnik Leningrad. Univ., 13, 15–23 (1960).Google Scholar
  80. 80.
    O. V. Pochinka, “Classification of Morse–Smale diffeomorphisms on 3-manifolds,” Dokl. Akad. Nauk, 440, No. 6, 34–37 (2011).zbMATHGoogle Scholar
  81. 81.
    A. Prishlyak, “Morse–Smale vector fields without closed trajectories on three-dimensional manifolds,” Mat. Zametki, 71, No. 2, 230–235 (2002).MathSciNetzbMATHGoogle Scholar
  82. 82.
    K. Sasano, “Links of closed orbits of non-singular Morse–Smale flows,” Proc. Amer. Math. Soc., 88, 727–734 (1983).MathSciNetzbMATHGoogle Scholar
  83. 83.
    M. Shub, “Morse–Smale diffeomorphisms are unipotent on homology,” Dynamical Systems, Academic Press, New York, 489–491 (1973).Google Scholar
  84. 84.
    M. Shub and D. Sullivan, “Homology theory and dynamical systems,” Topology, 4, 109–132 (1975).MathSciNetzbMATHGoogle Scholar
  85. 85.
    Ya.G. Sinay, “Markov partitions and U-diffeomorphisms,” Funktsional. Anal. i Prilozhen., 2, No. 1, 64–89 (1968).MathSciNetGoogle Scholar
  86. 86.
    Ya.G. Sinay, “Construction of Markov partitions,” Funkts. analiz i ego prilozh., 2, No. 3, 70–80 (1968).Google Scholar
  87. 87.
    S. Smale, “Morse inequalities for a dynamical system,” Bull. Amer. Math. Soc., 66, 43–49 (1960).MathSciNetzbMATHGoogle Scholar
  88. 88.
    S. Smale, “Generalized Poincare’s conjecture in dimensions greater than four,” Bull. Amer. Math. Soc., 66, 485–488 (1960).MathSciNetGoogle Scholar
  89. 89.
    S. Smale, “On gradient dynamical systems,” Ann. of Math. (2), 74, 199–206 (1961).MathSciNetzbMATHGoogle Scholar
  90. 90.
    S. Smale, “Generalized Poincare’s conjecture in dimensions greater than four,” Ann. of Math. (2), 74, 391–406 (1961).MathSciNetzbMATHGoogle Scholar
  91. 91.
    S. Smale, “Diffeomorphisms with many periodic points,” Differ. and Combinat. Topology, Sympos., Marston Morse, Princeton, 63–80 (1965).Google Scholar
  92. 92.
    S. Smale, “Differentiable dynamical systems,” Uspekhi Mat. Nauk, 25, No. 1, 113–185 (1970).MathSciNetzbMATHGoogle Scholar
  93. 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. 94.
    M. Wada, “Closed orbits of non-singular Morse–Smale flows on S 3,” J. Math. Soc. Japan, 41, 405–413 (1989).MathSciNetzbMATHGoogle Scholar
  95. 95.
    X. Wang, “The C *-algebras of Morse–Smale flows on two-manifolds,” Ergodic Theory Dynam. Systems, 10, 565–597 (1990).MathSciNetzbMATHGoogle Scholar
  96. 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).MathSciNetzbMATHGoogle Scholar
  97. 97.
    E.V. Zhuzhoma and V. S. Medvedev, “Morse–Smale systems with three nonwandering points,” Dokl. Akad. Nauk, 440, No. 1, 11–14 (2011).MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  • V. Z. Grines
    • 1
    • 2
    Email author
  • Ye. V. Zhuzhoma
    • 1
  • O. V. Pochinka
    • 1
  1. 1.National Research University “Higher School of Economics” in Nizhniy NovgorodNizhniy NovgorodRussia
  2. 2.Lobachevski State University of Nizhniy NovgorodNizhniy NovgorodRussia

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