Geodesic
On the Simple Isotopy Class of a Source--Sink Diffeomorphism on the $3$-Sphere
Matematičeskie zametki, Tome 94 (2013) no. 6, pp. 828-845

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The results obtained in this paper are related to the Palis–Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse–Smale systems on a closed smooth manifold $M^n$. Newhouse and Peixoto showed that such an arc joining flows exists for any $n$ and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For $n=1$, this is related to the presence of the Poincaré rotation number, and for $n=2$, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension $n=3$, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse–Smale diffeomorphism on the $3$-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.
Keywords: isotopic diffeomorphisms, Morse–Smale diffeomorphism, source-sink diffeomorphism, wildly embedded separatrices
Mots-clés : simple arc. 
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     author = {V. Z. Grines and O. V. Pochinka},
     title = {On the {Simple} {Isotopy} {Class} of a {Source--Sink} {Diffeomorphism} on the $3${-Sphere}},
     journal = {Matemati\v{c}eskie zametki},
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V. Z. Grines; O. V. Pochinka. On the Simple Isotopy Class of a Source--Sink Diffeomorphism on the $3$-Sphere. Matematičeskie zametki, Tome 94 (2013) no. 6, pp. 828-845. http://geodesic.mathdoc.fr/item/MZM_2013_94_6_a3/