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. 
@article{TM_2007_256_a2,
     author = {C. Bonatti and V. Z. Grines and V. S. Medvedev and O. V. Pochinka},
     title = {Bifurcations of {Morse--Smale} {Diffeomorphisms} with {Wildly} {Embedded} {Separatrices}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {54--69},
     publisher = {mathdoc},
     volume = {256},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2007_256_a2/}
}
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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/