Rough diffeomorphisms with basic sets of codimension one

The review is devoted to the exposition of results (including those of the authors of the review) obtained from the 2000s until the present, on topological classification of structurally stable cascades defined on a smooth closed manifold $M^n$ ($n\ge3$) assuming that their nonwandering sets either contain an orientable expanding (contracting) attractor (repeller) of codimension one or completely consist of basic sets of codimension one. The results presented here are a natural continuation of the topological classification of Anosov diffeomorphisms of codimension one. The review also reflects progress related to construction of the global Lyapunov function and the energy function for dynamical systems on manifolds (in particular, a construction of the energy function for structurally stable $3$-cascades with a nonwandering set containing a two-dimensional expanding attractor is described).