On Morse-Smale diffeomorphisms on manifolds of dimension higher than three

Oriented Peixoto graph, with a given permutation on its vertices, has been analyzed and theoretically proved to be a complete topological invariant for dimensions higher than three. The proof of this result essentially uses facts of multidimensional topology, and the results obtained in three or four dimension implies that the closures of invariant manifolds of diffeomorphisms from the class of Morse-Smale diffeomorphisms with higher dimensions greater than three cannot be nonlinearly embedded. A scheme for constructing the conjugating homeomorphism and also mention the fundamental difficulties was provided. The conditions defining the Morse-Smale diffeomorphisms class showed that, for any diffeomorphism, the nonwandering set consists of precisely one repulsive point and some periodic points.