Skip to main content Accessibility help

Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds

  • CH. BONATTI (a1), V. GRINES (a2), F. LAUDENBACH (a3) and O. POCHINKA (a2)


We show that, up to topological conjugation, the equivalence class of a Morse–Smale diffeomorphism without heteroclinic curves on a $3$ -manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.



Hide All
[1] Alexander, J.. On the subdivision of 3-spaces by polyhedron. Proc. Natl. Acad. Sci. USA 10 (1924), 68.
[2] Andronov, A. and Pontryagin, L.. Rough systems. Dokl. Akad. Nauk SSSR 14(5) (1937), 247250.
[3] de Baggis, G.. Rough systems of two differential equations. Uspekhi Mat. Nauk 10(4) (1955), 101126.
[4] Bing, R.. The Geometric Topology of 3-Manifolds (Colloquium Publications, 40) . American Mathematical Society, Providence, RI, 1983.
[5] Bonatti, Ch. and Grines, V.. Knots as topological invariant for gradient-like diffeomorphisms of the sphere S 3 . J. Dyn. Control Sys. 6(4) (2000), 579602.
[6] Bonatti, Ch., Grines, V., Medvedev, V. and Pecou, E.. Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves. Topol. Appl. 117 (2002), 335344.
[7] Bonatti, Ch., Grines, V., Medvedev, V. and Pecou, E.. Topological classification of gradient-like diffeomorphisms on 3-manifolds. Topology 43 (2004), 369391.
[8] Bonatti, Ch., Grines, V. and Pochinka, O.. Classification of the Morse–Smale diffeomorphisms with the finite set of heteroclinic orbits on 3-manifolds. Tr. Mat. Inst. Steklova 250 (2005), 553.
[9] Bonatti, Ch., Grines, V. and Pochinka, O.. Classification of Morse–Smale diffeomorphisms with the chain of saddles on 3-manifolds. Foliations 2005. World Scientific, Singapore, 2006, pp. 121147.
[10] Candel, A.. Laminations with transverse structure. Topology 38(1) (1999), 141165.
[11] Grines, V., Medvedev, V., Pochinka, O. and Zhuzhoma, E.. Global attractors and repellers for Morse–Smale diffeomorphisms. Tr. Mat. Inst. Steklova 271 (2010), 111133.
[12] Grines, V. and Pochinka, O.. Morse–Smale cascades on 3-manifolds. Russian Math. Surveys 68(1) (2013), 117173.
[13] Grines, V., Medvedev, T. and Pochinka, O.. Dynamical Systems on 2- and 3-Manifolds. Springer, Cham, 2016.
[14] Grines, V., Zhuzhoma, E. and Medvedev, V.. New relations for Morse–Smale systems with trivially embedded one dimensional separatrices. Sb. Math. 194(7) (2003), 9791007.
[15] Leontovich, E.. Some Mathematical Works of Gorky School of A. A. Andronov (Proceedings of the Third All-Union Mathematical Congress, vol. III). Akad. Nauk SSSR, Moscow, 1958, pp. 116125.
[16] Mayer, A.. Rough map circle to circle. Uch. Zap. GGU 12 (1939), 215229.
[17] Moise, E.. Geometric Topology in Dimensions 2 and 3 (Graduate Texts in Mathematics, 47) . Springer, New York, 1977.
[18] Palis, J.. On Morse–Smale dynamical systems. Topology 8(4) (1969), 385404.
[19] Palis, J. and de Melo, W.. Geometrical Theory of Dynamical Systems. Springer, New York, 1982.
[20] Palis, J. and Smale, S.. Structural Stability Theorems (Proceedings of the Institute on Global Analysis, 14) . American Mathematical Society, Providence, RI, 1970, pp. 223231.
[21] Peixoto, M.. On structural stability. Ann. of Math. (2) 69(1) (1959), 199222.
[22] Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1(2) (1962), 101120.
[23] Peixoto, M.. Structural stability on two-dimensional manifolds: a further remark. Topology 2(2) (1963), 179180.
[24] Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747817.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed