Interrelation Between the Dynamics of Morse–Smale Diffeomorphisms and the Topology of the Ambient 3-Manifold

  • Viacheslav Z. GrinesEmail author
  • Timur V. Medvedev
  • Olga V. Pochinka
Part of the Developments in Mathematics book series (DEVM, volume 46)


In this chapter we state some interrelations between the topology of the ambient manifold \(M^3\) and dynamics of a diffeomorphism \(f\in MS(M^3)\). These relation deal with the number
where \(r_{_f}\) is the number of the saddle periodic points and \(l_{_f}\) is the number of the knot periodic points of the diffeomorphism f. In the section 6.1 we construct the topological classification of closed orientable 3-manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves. Such a manifold is either the 3-sphere if \(g_{_f}=0\) or the connected sum of \(g_{_f}\) copies of \(\mathbb S^2\times \mathbb S^1\). We point out one more interrelation between \(g_{_f}\) and the topology of the manifold \(M^3\) if the diffeomorphism f is gradient-like and it has tamely embedded frames of 1-dimensional separatrices. In this case the ambient manifold \(M^3\) admits Heegaard splitting of genus \(g_{_f}\). The results of this chapter are for the most part contained in [1, 2].


Periodic Point Topological Embedding Relation Deal Invariant Frame Stable Separatrices 
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  1. 1.
    Bonatti, C., Grines, V., Medvedev, V., Pécou, E.: Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves. Topol. Appl. 117(3), 335–344 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Grines, V., Zhuzhoma, E., Medvedev, V.: New relations for Morse–Smale systems with trivially embedded one-dimensional separatrices. Sb. Math. 194(7), 979–1007 (2003). doi: 10.1070/SM2003v194n07ABEH000751 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Pochinka, O.: Diffeomorphisms with mildly wild frame of separatrices. Univ. Iagel. Acta Math. 47, 149–154 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viacheslav Z. Grines
    • 1
    Email author
  • Timur V. Medvedev
    • 2
    • 3
  • Olga V. Pochinka
    • 1
  1. 1.Department of Fundamental MathematicsNational Research University Higher School of EconomicsNizhny NovgorodRussia
  2. 2.Department of Differential Equations, Mathematical and Numerical AnalysisNizhny Novgorod State UniversityNizhny NovgorodRussia
  3. 3.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussia

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