On the Realization of Morse–Smale Diffeomorphisms with Heteroclinic Curves on a 3-Sphere

A Morse–Smale diffeomorphism is constructed on a three-dimensional sphere whose nonwandering set consists of one sink, one source, and two saddle fixed points. The two-dimensional manifolds of the saddle fixed points intersect along a unique one-dimensional heteroclinic curve. This example is constructed so that the one-dimensional separatrices of the saddle fixed points may represent wildly embedded arcs, which results in the realization of at least two topologically nonconjugate diffeomorphisms of the type under consideration. The example constructed shows an essential difference between the behavior of discrete dynamical systems on three-dimensional manifolds and analogous systems with continuous time (flows).