On the obstructions to the existence of a simple arc joining the multidimensional Morse-Smale diffeomorphisms.
In this paper we consider Morse-Smale diffeomorphisms defined on a multiply connected closed manifold M^n, n>3. For such systems, the concept of trivial (nontrivial) connectedness of their periodic orbits is introduced. It is established that isotopic trivial and non-trivial diffeomorphisms can not be joined by an arc with bifurcations of codimension one. Examples of such Morse-Smale cascades on a manifold S^(n-1)xS^1.