Necessary and sufficient conditions for topological classification of Morse–Smale cascades on 3-manifolds

In this paper class $MS(M^3)$ of Morse–Smale diffeomorphisms (cascades) given on connected closed orientable $3$-manifolds are considered. For a diffeomorphism $f\in MS(M^3)$ it is introduced a notion scheme $S_f$, which contains an information on the periodic data of the cascade and a topology of embedding of the sepsrstrices of the saddle points. It is established that necessary and sufficient condition for topological conjugacy of diffeomorphisms $f,f'\in MS(M^3)$ is the equivalence of the schemes $S_f$, $S_{f'}$.