• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Article

О включении диффеоморфизмов Морса-Смейла в топологический поток

This review presents the results of recent years on solving of the J. Palis's 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 diffeomophisms 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   recently  obtaining the  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.