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Published in:
2016 | OriginalPaper | Chapter
6. Interrelation Between the Dynamics of Morse–Smale Diffeomorphisms and the Topology of the Ambient 3-Manifold
Authors : Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka
Published in: Dynamical Systems on 2- and 3-Manifolds
Publisher: Springer International Publishing
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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 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].
$$g_{_f}=\frac{r_{_f}-l_{_f}+2}{2},$$