Topological Conjugacy of Gradient-Like Flows on Surfaces and Efficient Algorithms for Its Distinguition

Cover Page

Cite item


Gradient-like flows on surfaces have simple dynamics, which inspired many mathematicians to search for invariants of their topological equivalence. Under assumptions of different generality on the class of gradient-like flows under consideration, such classical invariants as the Leontovich- Mayer scheme, the Peixoto graph, the equipped Peixoto graph, the two-color Wang graph, the threecolor Oshemkov-Sharko graph, the Fleitas circular scheme, etc. were obtained. Thus, the problem of classifying gradient-like flows on surfaces from the point of view of topological equivalence has been solved in an exhaustive way. In recent works by Kruglov, Malyshev, and Pochinka, it was proved that for gradient-like flows the topological equivalence classes coincide with the topological conjugacy classes. The obtained result allows us to use any invariants of their equivalence for topological conjugacy of gradient-like flows. The present study is a review of the results on topological conjugacy of gradient-like flows on surfaces and efficient algorithms for its distinguishing, that is, algorithms whose running time is limited by some polynomial on the length of the input information.

About the authors

V. E. Kruglov

HSE University

Author for correspondence.
Nizhny Novgorod, Russia

O. V. Pochinka

HSE University

Nizhny Novgorod, Russia


  1. Андронов А.А., Понтрягин Л.С. Грубые системы// Докл. АН СССР. -1937.- 14, № 5.- С. 247-250.
  2. Леонтович Е.А., Майер А.Г. О траекториях, определяющих качественную структуру разбиения сферы на траектории// Докл. АН СССР. - 1937.- 14, № 5.-С. 251-257.
  3. Леонтович Е.А., Майер А.Г. О схеме, определяющей топологическую структуру разбиения на траектории// Докл. АН СССР. -1955.- 103, № 4.-С. 557-560.
  4. Майер А.Г. Грубые преобразования окружности// Уч. зап. ГГУ. -1939.- 12.- С. 215-229.
  5. Ошемков А.А., Шарко В.В. О классификации потоков Морса-Смейла на двумерных многообразиях// Мат. сб.- 1998.-189, № 8.-C. 93-140.
  6. Палис Ж., Ди Мелу В. Геометрическая теория динамических систем. Введение.- М.: Мир, 1986.
  7. Fleitas G. Classification of gradiet-like flows on dimensions two and three// Bol. Soc. Bras. Mat.- 1975.- 6.- С. 155-183.
  8. Grines V., Medvedev T., Pochinka O. Dynamical Systems on 2- and 3-Manifolds.-Cham: Springer, 2016.
  9. Hopcroft J.E., Wong J.K. Linear time algorithm for isomorphism of planar graphs: preliminary report// В сб.: «Proc. of the 6th Annual ACM Symposium on Theory of Computing». -Seattle, 1974.- С. 172-184.
  10. Kruglov V. Topological conjugacy of gradient-like flows on surfaces// Динам. сист.- 2018.- 8, № 1.- С. 15-21.
  11. Kruglov V., Malyshev D., Pochinka O. On algorithms that effectively distinguish gradient-like dynamics on surfaces// Arnold Math. J.-2018.- 4, № 3-4.- С. 483-504.
  12. Miller G. Isomorphism testing for graphs of bounded genus// В сб.: «Proceedings of the 12th Annual ACM Symposium on Theory of Computing».- New York: The Association for Computing Machinery, 1980.- С. 225-235.
  13. Peixoto M. Structural stability on two-dimensional manifolds// Topology.-1962.- 1, № 2.-С. 101-120.
  14. Peixoto M. Structural stability on two-dimensional manifolds (a further remarks)// Topology.- 1963.- 2, № 2. -С. 179-180.
  15. Peixoto M.M. On the classification of flows on 2-manifolds// В сб.: «Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971».-1973.-С. 389-419.
  16. Pugh C., Shub M., The Ω-stability theorem for flows// Invent. Math.- 1970.- 11, № 2.-С. 150-158.
  17. Robinson C. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos.- Boca Raton: CRC Press, 1999.
  18. Smale S. Differentiable dynamical systems// Bull. Am. Math. Soc.- 1967.- 73.-С. 747-817.
  19. Wang X. The C∗-algebras of Morse-Smale flows on two-manifolds// Ergodic Theory Dynam. Sytems.- 1990.-10.-С. 565-597.

Copyright (c) 2022 Contemporary Mathematics. Fundamental Directions

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

This website uses cookies

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

About Cookies