On the Simple Isotopy Class of a Source–Sink Diffeomorphism on the $3$-Sphere

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.