Geodesic
Bifurcations of Morse--Smale Diffeomorphisms with Wildly Embedded Separatrices
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 54-69.

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We study bifurcations of Morse–Smale diffeomorphisms under a change of the embedding of the separatrices of saddle periodic points in the ambient 3-manifold. The results obtained are based on the following statement proved in this paper: for the 3-sphere, the space of diffeomorphisms of North Pole–South Pole type endowed with the $C^1$ topology is connected. This statement is shown to be false in dimension 6. 
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     title = {Bifurcations of {Morse--Smale} {Diffeomorphisms} with {Wildly} {Embedded} {Separatrices}},
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C. Bonatti; V. Z. Grines; V. S. Medvedev; O. V. Pochinka. Bifurcations of Morse--Smale Diffeomorphisms with Wildly Embedded Separatrices. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 54-69. http://geodesic.mathdoc.fr/item/TM_2007_256_a2/

[1] Banyaga A., “Sur la structure du groupe des difféomorphismes qui preservent une forme symplectique”, Comment. Math. Helv., 53 (1978), 174–227 | DOI | MR | Zbl

[2] Belitskii G.R., Normalnye formy, invarianty i lokalnye otobrazheniya, Nauk. dumka, Kiev, 1979 | MR

[3] Bonatti C., Grines V.Z., “Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dyn. and Control Syst., 6:4 (2000), 579–602 | DOI | MR | Zbl

[4] Bonatti Ch., Grines V., Pochinka O., On existence of a smooth arc joining “North pole–South pole” diffeomorphisms, Prepubl. Inst. Math. Bourgogne, 2006 \href{http://math.u-bourgogne.fr/topologie/prepub/bifs_4.pdf} {http://math.u-bourgogne.fr/topol

[5] Cerf J., Sur les difféomorphismes de la sphère de dimension trois ($\Gamma_4=0$), Lect. Notes Math., 53, Springer, Berlin, 1968 | MR | Zbl

[6] Kirby R.C., Siebenmann L.C., Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Stud., 88, Princeton Univ. Press, Princeton (NJ), 1977 | MR | Zbl

[7] Laudenbach F., Topologie de la dimension trois: homotopie et isotopie, Astérisque, 12, Centre Math. Ecole Polytech., Paris, 1974 | MR

[8] Matsumoto S., “There are two isotopic Morse–Smale diffeomorphism which cannot be joined by simple arcs”, Invent. math., 51 (1979), 1–7 | DOI | MR | Zbl

[9] Milnor Dzh., “O mnogoobraziyakh, gomeomorfnykh semimernoi sfere”, Matematika, 1:3 (1957), 35–42

[10] Milnor Dzh., Teorema ob $h$-kobordizme, Mir, M., 1969 | MR

[11] Milnor Dzh., Uolles A., Differentsialnaya topologiya: Nachalnyi kurs, Mir, M., 1972 | MR | Zbl

[12] Newhouse S., Peixoto M.M., “There is a simple arc joining any two Morse–Smale flows”, Astérisque, 31, 1976, 15–41 | MR

[13] Palis Zh., di Melu V., Geometricheskaya teoriya dinamicheskikh sistem, Mir, M., 1986 | MR

[14] Palis J., Pugh C.C., “Fifty problems in dynamical systems”, Lect. Notes Math., 468, 1975, 345–353 | MR | Zbl

[15] Pixton D., “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172 | DOI | MR | Zbl

[16] Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L., Metody kachestvennoi teorii v nelineinoi dinamike, ch. 1, In-t kompyut. issled., Moskva–Izhevsk, 2004