Dynamical Systems on 2- and 3-Manifolds pp 119-130 | Cite as

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

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## Abstract

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

$$g_{_f}=\frac{r_{_f}-l_{_f}+2}{2},$$

*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].## Keywords

Periodic Point Topological Embedding Relation Deal Invariant Frame Stable Separatrices
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 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.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.Pochinka, O.: Diffeomorphisms with mildly wild frame of separatrices. Univ. Iagel. Acta Math.
**47**, 149–154 (2009)MathSciNetzbMATHGoogle Scholar

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