4. Wild Embedding of the Separatrices into 3-Manifolds and Pixton Diffeomorphism

The Morse-Smale diffeomorphisms on 3-manifolds are in marked contrast to the Morse-Smale flows and diffeomorphisms on 2-manifolds because of the possibility of wild embedding of the separatrices of the saddle points. In this chapter we prove the criteria of the tame embedding of 1- and 2-dimensional separatrices. We give the complete study of the simplest Morse-Smale diffeomorphisms with the wandering set of four points if exactly one of them is the saddle (the Pixton diffeomorphisms). We show that the invariant manifolds of the saddle point of such diffeomorphisms can be wildly embedded. It implies that there are countably many topologically non-conjugate simplest diffeomorphisms. The topological invariant in this case is the knot in the characteristic space homeomorphic to \(\mathbb S^2\times \mathbb S^1\). Then we study the bifurcations through which the transition from one class of topologically conjugate diffeomorphisms to another occurs. The distinctive specialty of the new bifurcation is that the structure on the non-wandering set does not change but the qualitative change of the diffeomorphism is due to the change of the type of the embedding of the separatrices of the saddle points. This problem is connected to the problem of J. Palis and C. Pugh [8], that is to find a smooth curve with some “good” properties (finitely many bifurcations for instance) that joins two structurally stable systems (flows or diffeomorphisms). S. Newhouse and M. Peixoto showed in [7] that any two Morse-Smale flows on a closed manifold can be joined by an arc with finitely many bifurcations. Discrete systems are in contrast to this result. For example, S. Matsumoto showed in [5] that every orientable closed surface admits two isotopic Morse-Smale diffeomorphisms which cannot be joined by such an arc. For the dimensions larger or equal to 3 the problem is nontrivial even for the simplest diffeomorphisms “north pole - south pole”. The classical result of J. Cerf [3] states that for every two orientation preserving diffeomorphisms (and therefore for every two diffeomorphisms “north pole - south pole”) on \(\mathbb {S}^3\) there is a smooth arc joining them. We show that it can be so chosen that the whole arc consists of the “north pole - south pole” diffeomorphisms. We show that for dimensions larger then 3 the problem is even more complicated. On the sphere \(\mathbb S^6\) there are two “north pole - south pole” diffeomorphisms that cannot be joined by a smooth arc (it follows easily from the result by J. Milnor). Finally we show the way in which the sequence of two saddle-knot bifurcations results in transition from one class of topological conjugacy to another in set of Pixton diffeomorphisms. The main results of this chapter are in the papers [1, 2, 4, 9].