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Article

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

Mathematical notes. 2013. Vol. 94. No. 6. P. 862-875.

The results obtained in this paper are related to the Palis–Pugh problem on the existence of an arc withfinitely or countably many bifurcations which joins two Morse–Smale systems on a closed smooth manifoldMn . Newhouse and Peixoto showed that such an arc joining flows exists for any nand, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. Forn=1, this is related to the presence of the Poincar´ erotationnumber, and forn=2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimensionn=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.