Date: 12 Apr 2011

On Topological Classification of Morse–Smale Diffeomorphisms

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Abstract

Well-known results on topological classification of Morse–Smale flows were obtained by Leontovich and Maier (Dokl Akad Nauk 103(4):557–560, 1955) for flows on the two dimensional sphere, and by Mauricio Peixoto (Ann Math 69:199–222, 1959; On a classification of flows on 2-manifolds. Proc. Symp. Dyn. Syst. Salvador 389–492, 1973) for flows on any closed surfaces. Since 1980s rather great progress was achieved in classification of Morse–Smale diffeomorphisms on surfaces. For such diffeomorphisms with finite number heteroclinic orbits there is complete invariant in the form of a graph (similar to that introduced by Peixoto for flows). This graph is defined by taking into account the heteroclinic intersections and it is equipped with a graph automorphism induced by the given diffeomorphism (see for example the surveys (Bonatti et al. Comput Appl Math 20(1–2):11–50, 2001; Grines J Dyn Control Syst 6(1):97–126, 2000) for references and details). Describing of Morse–Smale diffeomorphisms with infinite set of heterolinic orbits uses Markov chains endowed by additional information (see Bonatti et al. Comput Appl Math 20(1–2):11–50, 2001). A progress in dimension 3 is based on rather recent results on finding new topological knot and link invariants which describe (possibly, wild) embedding of invariant manifolds of saddle periodic points into the ambient manifold. These invariants allowed to discover a principal distinctive phenomenon of Morse–Smale diffeomorphisms in dimension 3: the existence of a countable set of non-conjugate Morse–Smale diffeomorphisms with isomorphic Peixoto graphs.140pt]The first author “Viacheslav Grines" has been considered as corresponding author. Please check. The main goal of the present survey is to give an exposition of recent results on classification of Morse–Smale diffeomorphis on 3-manifolds.