Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices

Abstract

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.