Classification of Morse-Smale diffeomorphisms on 3-manifolds
type: Item
instance of: scholarly article
title: Classification of Morse-Smale diffeomorphisms on 3-manifolds
zbMATH Open document ID: 1260.37014
DOI: 10.1134/S106456241106038X
published in: Doklady Mathematics
publication date: 2012-01-06T00:00:00Z
review text: 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 abs…
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 37D10
zbMATH DE Number: 5995575
zbMATH Keywords: Morse-Smale diffeomorphism
zbMATH Keywords: 3-manifold
author: O. V. Pochinka
reviewed by: David E. Hurtubise
MaRDI profile type: Publication
cites work: Knots as topological invariants for gradient-like diffeomorphisms of the sphere \(S^3\)
cites work: 3-manifolds which are orbit spaces of diffeomorphisms
cites work: Wild unstable manifolds
cites work: Morse inequalities for a dynamical system
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description: scientific article; zbMATH DE number 5995575