Duke Mathematical Journal

Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

C. Bonatti, V. Grines, and O. Pochinka

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Abstract

The topological classification of even the simplest Morse–Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possibility of “wild” behavior of separatrices of saddle points. Another difference between Morse–Smale diffeomorphisms in dimension 3 and their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may not be only a point, as in the 2-dimensional case, but also a curve, compact or noncompact. The problem of topological classification of Morse–Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2016 by C. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, and O. Pochinka. The present article is devoted to completing the topological classification of the set MS(M3) of orientation-preserving Morse–Smale diffeomorphisms f on a smooth closed orientable 3-manifold M3. The complete topological invariant for a diffeomorphism fMS(M3) is the equivalence class of its scheme Sf which contains information on the periodic data and the topology of embedding of 2-dimensional invariant manifolds of the saddle periodic points of f into the ambient manifold.

Article information

Source
Duke Math. J., Volume 168, Number 13 (2019), 2507-2558.

Dates
Received: 18 October 2017
Revised: 24 February 2019
First available in Project Euclid: 7 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1567821623

Digital Object Identifier
doi:10.1215/00127094-2019-0019

Mathematical Reviews number (MathSciNet)
MR4007599

Subjects
Primary: 37B25: Lyapunov functions and stability; attractors, repellers
Secondary: 37D15: Morse-Smale systems 57M30: Wild knots and surfaces, etc., wild embeddings

Keywords
Morse–Smale diffeomorphism topological classification

Citation

Bonatti, C.; Grines, V.; Pochinka, O. Topological classification of Morse–Smale diffeomorphisms on $3$ -manifolds. Duke Math. J. 168 (2019), no. 13, 2507--2558. doi:10.1215/00127094-2019-0019. https://projecteuclid.org/euclid.dmj/1567821623


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