On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

Cover Page

Cite item

Full Text

Abstract

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Palis’s problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

About the authors

V. Z. Grines

National Research University “Higher School of Economics”

Author for correspondence.
Email: vgrines@yandex.ru
Nizhniy Novgorod

E. Ya. Gurevich

National Research University “Higher School of Economics”

Email: egurevich@hse.ru
Nizhniy Novgorod

O. V. Pochinka

National Research University “Higher School of Economics”

Email: opochinka@yandex.ru
Nizhniy Novgorod

References

  1. Бонатти Хр., Гринес В. З., Починка О. В. Классификация диффеоморфизмов Морса-Смейла с конечным множеством гетероклинических орбит на 3-многообразиях// Докл. АН СССР. - 2004. - 396, № 4. - C. 439-442. С. 439-442
  2. Бонатти Х., Гринес В. З., Починка О. В. Реализация диффеоморфизмов Морса-Смейла на 3многообразиях// Тр. МИАН. - 2017. - 297. - C. 46-61. 297. С. 35-49.
  3. Брин М. И. О включении диффеоморфизма в поток// Изв. вузов. Сер. Мат. - 1972. - 8.- C. 19-25.
  4. Гринес В. З., Гуревич Е. Я., Медведев В. С. Граф Пейкшото диффеоморфизмов Морса-Смейла на многообразиях размерности большей трех// Тр. МИАН. - 2008. - 261. - C. 61-86. (2008), 59-83
  5. Гринес В. З., Гуревич Е. Я., Медведев В. С. О топологической классификации диффеоморфизмов Морса-Смейла с одномерным множеством неустойчивых сепаратрис на многообразиях размерности большей 3// Тр. МИАН. - 2010. - 270. - C. 62-86. 270 (2010), 57-79
  6. Гринес В. З., Гуревич Е. Я., Починка О. В., Медведев В. С. О включении в поток диффеоморфизмов Морса-Смейла на многообразиях размерности, большей двух// Мат. заметки. - 2012. - 91, № 5. - С. 791-794. Notes, 91:5 (2012), 742-745
  7. Гринес В. З., Жужома Е. В., Медведев В. С., Починка О. В. Глобальные аттрактор и репеллер диффеоморфизмов Морса-Смейла// Тр. МИАН. - 2010. - 271. - C. 111-133. 103-124
  8. Гробман Д. М. О гомеоморфизме систем дифференциальных уравнений// Докл. АН СССР. - 1959. - 128, № 5. - 1959. - C. 880-881.
  9. Гробман Д. М. Топологическая классификация окрестностей особой точки в n-мерном пространстве// Мат. сб. - 1962. 56, № 1. - С. 77-94.
  10. Гуревич В., Волмэн Г. Теория размерности. - М.: Изд-во иностранной литературы, 1948.
  11. Жужома Е. В., Медведев В. С. Непрерывные потоки Морса-Смейла с тремя состояниями равновесия// Мат. сб. - 2016. - 207, № 5. - С. 69-92.
  12. Пилюгин С. Ю. Фазовые диаграммы, определяющие системы Морса-Смейла без периодических траекторий на сферах// Дифф. уравн. - 1978. - 14, № 2. - C. 245-254.
  13. Починка О. В., Гринес В. З., Гуревич Е. Я., Медведев В. С. О включении диффеоморфизмов Морса- Смейла на 3-многообразии в топологический поток// Мат. сб. - 2012. - 203, № 12. - С. 81-104. 1761- 1784
  14. Artin E., Fox R. H. Some wild cells and spheres in three-dimensional space// Ann. Math. - 1948. - 49.- С. 979-990.
  15. Blankinship W. A. Generalization of a construction of Antoine// Ann. Math. - 1951. - 2, № 3. - C. 276- 297.
  16. Bonatti Ch., Grines V. Knots as topological invariant for gradient-like diffeomorphisms of the sphere S3// J. Dyn. Control Syst. - 2000. - 6, № 4. - С. 579-602.
  17. Bonatti C., Grines V., Laudenbach F., Pochinka O. Topological classification of Morse-Smale diffeomorphisms without heteroclinic curves on 3-manifolds// Ergodic Theory Dynam. Systems. - 2019. - 39, №. 9. - С. 2403-2432.
  18. Bonatti Ch., Grines V., Medvedev V., Pe´ cou E. Topological classification of gradient-like diffeomorphisms on 3-manifolds// Topology.- 2004.- 43. - С. 369-391.
  19. Bonatti C., Grines V., Pochinka O. Topological classification of Morse-Smale diffeomorphisms on 3manifolds// Duke Math. J. - 2019. - 168, № 13. - С. 2507-2558.
  20. Brown M. Locally flat imbeddings of topological manifolds// Ann. Math. (2). - 1962. - 75, № 2. - С. 331- 341.
  21. Cantrell J. C. Almost locally flat embeddings of Sn-1 in Sn// Bull. Am. Math. Soc. - 1963. - 69.- С. 716-718.
  22. Cantrell J. C. Almost locally poliedral curves in Euclidean n-space// Trans. Am. Math. Soc. - 1963. - 107, № 3. - С. 451-457.
  23. Cantrell J. C. n-frames in Euclidean k-space// Proc. Am. Math. Soc. - 1964. - 15, № 4. - С. 574-578.
  24. Chernavskii A. V. Piecewise linear approximation of imbeddings of manifolds in codimensions greater than two// Sb. Math. - 1970. - 11, № 3. - С. 465-466.
  25. Daverman R. J. Embeddings of (n - 1)-spheres in Euclidean n-space// Bull. Am. Math. Soc. - 1978. - 84, № 3. - С. 377-405.
  26. Debruner H., Fox R. A mildly wild embedding of an n-frame// Duke Math. J. - 1960. - 27, № 3. - С. 425-429.
  27. Dugundji J., Antosiewicz H. A. Parallelizable flows and Lyapunov’s second method// Ann. Math. - 1961. - 2, № 73. - C. 543-555.
  28. Foland N. E., Utz W. R. The embedding of discrete flows in continuous flows// В сб.: «Ergodic theory», Proc. Int. Symp., Tulane University, New Orleans, USA, October, 1961. - New York: Academic Press, 1963. - С. 121-134.
  29. Garay B. M. Discretization and some qualitative properties of ordinary differential equations about equilibria// Acta Math. Univ. Comenian. (N.S.). - 1993. - 62, № 2. - С. 249-275.
  30. Garay B. M. On structural stability of ordinary differential equations with respect to discretization methods// Numer. Math. - 1996. - 72, № 4. - С. 449-479.
  31. Grines V., Gurevich E., Pochinka O. Topological classification of Morse-Smale diffeomorphisms without heteroclinic intersections// J. Math. Sci. (N.Y.). - 2015. - 208, № 1. - С. 81-90.
  32. Grines V., Gurevich E., Pochinka O. On embedding of multidimensional Morse-Smale diffeomorphisms in topological flows// Mosc. Math. J. - 2019. - 19, № 4. - С. 739-760.
  33. Grines V., Gurevich E., Pochinka O. On topological classification of Morse-Smale diffeomorphisms on the sphere Sn// ArXiv. - 2019. - 1911.10234v2 [math.DS].
  34. Hartman P. On the local linearization of differential equations// Proc. Am. Math. Soc. - 1963. - 14, № 4. - С. 568-573.
  35. Hirsch M., Pugh C., Shub M. Invariant Manifolds. - Berlin-Heidelberg-New York: Springer-Verlag, 1977.
  36. Hudson J. F. Concordance and isotopy of PL embeddings// Bull. Am. Math. Soc. - 1966. - 72, № 3. - С. 534-535.
  37. Hudson J. F., Zeeman E. C. On combinatorial isotopy// Publ. IHES. - 1964. - 19. - С. 69-74.
  38. Kuperberg K. 2-wild trajectories// Discrete Contin. Dyn. Syst. - 2005. - Suppl. Vol. - С. 518-523.
  39. Max N. L. Homeomorphisms of Sn × S1// Bull. Am. Math. Soc. - 196. - 74, № 6. - С. 939-942.
  40. Medvedev T., Pochinka O. The wild Fox-Artin arc in invariant sets of dynamical systems// Dyn. Syst. - 2018. - 33, № 4. - С. 660-666.
  41. Miller R. T. Approximating codimension 3 embeddings// Ann. Math. (2). - 1972. - 95, № 3. - С. 406- 416.
  42. Palis J. On Morse-Smale dynamical systems// Topology. - 1969. - 8, № 4. - С. 385-404.
  43. Palis J. Vector fields generate few diffeomorphisms// Bull. Am. Math. Soc. - 1974. - 80. - С. 503-505.
  44. Palis J., Smale S. Structural stability theorem// В сб.: «Global Analysis», Proc. Symp. Pure Math., 1970, № 14. - Providence: American Math. Soc., 1970.
  45. Pixton D. Wild unstable manifolds// Topology. - 1977. - 16, № 2. - С. 167-172.
  46. Pochinka O. Diffeomorphisms with mildly wild frame of separatrices// Zesz. Nauk. Uniw. Jagiell. - 2009. - 47. - С. 149-154.
  47. Smale S. Differentiable dynamical systems// Bull. Am. Math. Soc. - 1967. - 73, № 6. - С. 747-817.
  48. Weller G. P. Locally flat imbeddings of topological manifolds in codimension three// Trans. Am. Math. Soc. - 1971. - 157. - С. 161-178.
  49. Young G. S. On the factors and fiberings of manifolds// Proc. Am. Math. Soc. - 1950. - 1. - С. 215-223.
  50. Zhuzhoma E. V., Medvedev V. S. Morse-Smale systems with few non-wandering points// Topology Appl. - 2013. - 160, № 3. - С. 498-507.

Copyright (c) 2020 Contemporary Mathematics. Fundamental Directions

License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies