Conjugacy of Smale-Vietoris diffeomorphisms using a conjugacy of endomorphisms

In the paper one gets the connection between a conjugacy of Smale-Vietoris diffeomorphisms and corresponding nonsingular circle endomorphisms conjugacy. To be precise, one gets the necessary condition for a conjugacy of the restriction of Smale-Vietoris diffeomorphism on basic manifolds. It is shown that a necessary condition for a conjugacy of the diffeomorphisms’ class under consideration is a conjugacy of corresponding circle endomorphisms. In the paper the technical theorem is also proved that contains some sufficient conditions of the existence of homeomorphism on basic manifolds that takes the orbits of Smale-Vietoris diffeomorphism to the orbits of another diffeomorphism with a commutative diagram of mappings. Together with the first result it gives the particular solution of the topological equivalence problem. Later on, the results of this paper may be useful to find invariants of conjugacy for diffeomorphisms of the class under consideration on basic manifolds.