Classification of Morse-Smale diffeomorphisms on 3-manifolds
Let \(M^3\) be an orientable \(3\)-manifold and \(MS(M^3)\) the set of orientation-preserving Morse-Smale diffeomorphisms on \(M\). In this paper, the author gives a complete topological classification of the conjugacy classes of \(MS(M^3)\) in terms of equivalence classes of schemes, which contain information on periodic data and the topology of the embeddings and intersections in the phase space of the \(2\)-dimensional invariant manifolds of saddle periodic points. \smallpagebreak Let \(f \in MS(M^3)\) and let \(\Omega_q\) denote the set of periodic points of \(f\) with Morse index \(q\) for \(q=0,1,2,3\). Let \(A_f = \Omega_0 \cup W^u_{\Omega_1}\), \(R_f = \Omega_3 \cup W^s_{\Omega_2}\) and \(V_f = M^3 - (A_f \cup R_f)\), where \(W^u_{\Omega_1}\) and \(W^s_{\Omega_2}\) denote unstable and stable manifolds respectively. The author proves that \(A_f\) is an attractor of \(f\), \(R_f\) is a repeller of \(f\), and \(V_f\) consists of wandering points which move to the attractor under the action of \(f\). \smallpagebreak Let \(\hat{V}_f = V_f/f\) denote the orbit space of the action, and let \(p_f:V_f \rightarrow \hat{V}_f\) denote the natural projection. The map \(p_f\) is a covering that induces an epimorphism \(\eta_f:\pi_1(\hat{V}_f) \rightarrow {\mathbb Z}\) and the orbit space \(\hat{V}_f\) is a simple manifold. Let \(\hat{{\mathbb W}}_f^s = p_f(W^s_{\Omega_1} - A_f)\) and \(\hat{{\mathbb W}}_f^u = p_f(W^u_{\Omega_2} - R_f)\). The set \(S_f = (\hat{V}_f, \eta_f, \hat{{\mathbb W}}_f^s, \hat{{\mathbb W}}_f^u)\) is called the scheme of the diffeomorphism \(f \in MS(M^3)\), and two schemes \(S_f\) and \(S_{f'}\) are said to be equivalent if there exists a homeomorphism \(\hat{\varphi}: \hat{V}_f \rightarrow \hat{V}_{f'}\) such that (i) \(\eta_f = \eta_{f'} \hat{\varphi}_\ast\) and (ii) \(\hat{\varphi} (\hat{{\mathbb W}}_f^s) = \hat{{\mathbb W}}_f^s\) and \(\hat{\varphi} (\hat{{\mathbb W}}_f^u) = \hat{{\mathbb W}}_f^u\). The author also defines the concept of an abstract scheme \(S = (\hat{V}, \eta, {\mathbb W}^s, {\mathbb W}^u)\) in terms of laminations and surgeries which is independent of any diffeomorphism. \smallpagebreak The main theorems of the paper can now be stated as follows. {Theorem 1.} Morse-Smale diffeomorphisms \(f,f' \in MS(M^3)\) are topologically conjugate if and only if their schemes are equivalent. { Theorem 2.} The scheme \(S_f\) of any diffeomorphism \(f \in MS(M^3)\) belongs to the set of abstract schemes. { Theorem 3.} For any abstract scheme \(S\) there exists a diffeomorphism \(f_S\in MS(M^3)\) whose scheme is equivalent to to \(S\).
scientific article; zbMATH DE number 5995575
scientific article