Duke Mathematical Journal

Topological classification of Morse–Smale diffeomorphisms on $3$-manifolds

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 $MS(M^{3})$ of orientation-preserving Morse–Smale diffeomorphisms $f$ on a smooth closed orientable $3$-manifold $M^{3}$. The complete topological invariant for a diffeomorphism $f\in MS(M^{3})$ is the equivalence class of its scheme $S_{f}$ 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
Revised: 24 February 2019
First available in Project Euclid: 7 September 2019

https://projecteuclid.org/euclid.dmj/1567821623

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

Mathematical Reviews number (MathSciNet)
MR4007599

Zentralblatt MATH identifier
07131293

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

References

• [1] R. H. Fox and E. Artin, Some wild cells and spheres in three-dimensional space Ann. of Math. (2) 49 (1948), 979–990.
• [2] A. N. Bezdenezhykh and V. Z. Grines, Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds, I, Selecta Math. Soviet. 11 (1992), 1–11.
• [3] A. N. Bezdenezhykh and V. Z. Grines, Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds, II, Selecta Math. Soviet. 11 (1992), 13–17.
• [4] C. Bonatti, S. Crovisier, G. Vago, and A. Wilkinson, Local density of diffeomorphisms with large centralizers, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 6, 925–954.
• [5] C. Bonatti and V. Grines, Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^{3}$, J. Dynam. Control Systems 6 (2000), no. 4, 579–602.
• [6] C. Bonatti, V. Grines, F. Laudenbach, and O. Pochinka, Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on $3$-manifolds, Ergodic Theory Dynam. Systems 39 (2019), no. 9, 2403–2432.
• [7] C. Bonatti, V. Grines, V. Medvedev, and E. Pecou, Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves, Topology Appl. 117 (2002), no. 3, 335–344.
• [8] C. Bonatti, V. Grines, V. Medvedev, and E. Pecou, Topological classification of gradient-like diffeomorphisms on $3$-manifolds, Topology 43 (2004), no. 2, 369–391.
• [9] C. Bonatti, V. Grines, and O. Pochinka, Classification of the Morse–Smale diffeomorphisms with a finite set of heteroclinic orbits on $3$-manifolds (in Russian), Tr. Mat. Inst. Steklova 250 (2005), 5–53; English translation in Proc. Steklov Inst. Math. 250 (2005), no. 3, 1–46.
• [10] C. Bonatti, V. Grines, and O. Pochinka, Realization of Morse–Smale diffeomorphisms on $3$-manifolds (in Russian), Tr. Mat. Inst. Steklova 297 (2017), 46–61; English translation in Proc. Steklov Inst. Math. 297 (2017), no. 1, 35–49.
• [11] C. Bonatti and R. Langevin, Difféomorphismes de Smale des surfaces, Astérisque 250, Soc. Math. France, Paris, 1998.
• [12] C. Bonatti and L. Paoluzzi, $3$-manifolds which are orbit spaces of diffeomorphisms, Topology 47 (2008), no. 2, 71–100.
• [13] C. E. Burgess, Embeddings of surfaces in euclidean three-space, Bull. Amer. Math. Soc. 81 (1975), no. 5, 795–818.
• [14] D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), no. 1, 83–107.
• [15] V. Grines, Topological classification of Morse–Smale diffeomorphisms with a finite set of heteroclinic trajectories on surfaces (in Russian), Mat. Zametki 54 (1993), no. 3, 3–17; English translation in Math. Notes 54 (1993), no. 3–4, 881–889.
• [16] V. Z. Grines, On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers, Mat. Sb. 188 (1997), no. 4, 537–569; English translation in Sb. Math. 188 (1997), no. 4–4, 537–569.
• [17] V. Z. Grines, E. Y. Gurevich, V. S. Medvedev, and O. V. Pochinka, On the embedding of Morse–Smale diffeomorphisms on a $3$-manifold in a topological flow (in Russian), Mat. Sb. 203 (2012), no. 12, 81–104; English translation in Sb. Math. 203 (2012), no. 11–12, 1761–1784.
• [18] V. Z. Grines, T. V. Medvedev, and O. V. Pochinka, Dynamical Systems on $2$- and $3$-Manifolds, Dev. Math. 46, Springer, Cham, 2016.
• [19] V. Z. Grines, V. S. Medvedev, and E. V. Zhuzhoma, On surface attractors and repellers on $3$-manifolds (in Russian), Mat. Zametki 78 (2005), no. 6, 813–826; English translation in Math. Notes 78 (2005), no. 5–6, 757–767.
• [20] V. Z. Grines and O. Pochinka, Morse–Smale cascades on $3$-manifolds (in Russian), Uspekhi Mat. Nauk 68 (2013), no. 1, 129–188; English translation in Russian Math. Surveys 68 (2013), 117–173.
• [21] V. Grines, E. Zhuzhoma, V. Medvedev, and O. Pochinka, Global attractor and repeller of Morse–Smale diffeomorphisms (in Russian), Tr. Mat. Inst. Steklova 271 (2010), 111–133; English translation in Proc. Steklov Inst. Math. 271 (2010), no. 1, 103–124.
• [22] O. G. Harrold, Jr., H. C. Griffith, and E. E. Posey, A characterization of tame curves in three-space, Trans. Amer. Math. Soc. 79 (1955), 12–34.
• [23] C. Kosniowsky, A First Course in Algebraic Topology, Cambridge Univ. Press, Cambridge, 1980.
• [24] E. Leontovich and A. Maier, On a scheme determining the topological structure of the separation of trajectories, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 557–560.
• [25] E. E. Moise, Geometric Topology in Dimensions $2$ and $3$, Grad. Texts in Math. 47, Springer, New York, 1977.
• [26] J. Palis, On Morse-Smale dynamical systems, Topology 8 (1969), 385–404.
• [27] J. Palis and W. de Melo, Geometrical Theory of Dynamical Systems, Springer, New York, 1982.
• [28] J. Palis and S. Smale, “Structural stability theorems” in Global Analysis (Berkeley, CA, 1968), Proc. Sympos. Pure Math. XIV, Amer. Math. Soc., Providence, 1970, 223–231.
• [29] M. Peixoto, “On the classification of flows on $2$-manifolds” in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973. 389–419.
• [30] D. Pixton, Wild unstable manifolds, Topology 16 (1977), no. 2, 167–172.
• [31] A. O. Prishlyak, Morse–Smale vector fields without closed trajectories on $3$-manifolds (in Russian), Mat. Zametki 71 (2002), no. 2, 254–260; English translation in Math. Notes 71 (2002), no. 1–2, 230–235.
• [32] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Stud. Adv. Math., 2nd ed., CRC Press, Boca Raton, FL, 1999.
• [33] D. Rolfsen, Knots and Links, Math. Lecture Ser. 7, Publish or Perish, Houston, 1990.
• [34] S. Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), no. 1, 199-206.
• [35] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.
• [36] W. P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, Princeton Math. Ser. 35, Princeton Univ. Press, Princeton, 1997.