On Morse-Smale diffeomorphisms on manifolds of dimension higher than three
Let \(G(M^n)\) be the set of orientation-preserving Morse-Smale diffeomorphisms on a connected closed orientable manifold \(M^n\) of dimension \(n>3\) such that, for any \(f\in G(M^n)\), the non-wandering set \(\Omega(f)\) of \(f\) has the following properties: (i) for all saddle periodic point \(\sigma\) from \(\Omega(f)\) the dimension of the unstable manifold \(W^u(\sigma)\) has dimension one, (ii) the unstable and stable manifolds of different saddle periodic points from \(\Omega (f)\) do not intersect. The authors try to obtain the topological classification of diffeomorphisms of \(G(M^n)\). Let \(\sigma\) be a saddle point of \(f\in G(M^n)\) and let \(W^s(\sigma)\) be the stable manifold of \(\sigma\). The connected components of \(W^\delta(\sigma)\setminus\sigma\) for \(\delta \in\{s,u\}\) are called the separatrices of \(\sigma\). To each diffeomorphism \(f\in G(M^n)\) the authors assign an oriented graph \(\Gamma(f)\) with vertex set isomorphic to the set \(\Omega(f)\) and edge set isomorphic to the set of separatices of saddle periodic points. Let \(P(f)\) be the permutation on the vertex set of \(\Gamma(f)\) induced by the restriction of \(f\) to the set of periodic points. We say that the graphs \(\Gamma(f)\) and \(\Gamma(f')\) of \(f,f'\in G(M^n)\) are isomorphic if there exists a graph isomorphism \(\eta:\Gamma(f)\to\Gamma(f')\) preserving the orientations of edges and such that \(P(f')=\eta P(f)\eta^{-1}\). The authors sketch the proof of the following theorem: Diffeomorphisms \(f,f'\in G(M^n)\) \((n>3)\) are topologically conjugate if and only if the graphs \(\Gamma(f)\) and \(\Gamma(f')\) are isomorphic.
topological classification of Morse-Smale diffeomorphisms
scientific article; zbMATH DE number 5375878