Skew products and geometrically integrable maps: results, problems and prospects

Skew products on the simplest manifolds of a finite dimension are considered. The fundamental theorem on the decomposition of the space of $C^1$-smooth skew products into the finite union of subspaces is proved. The theorem is explained for the case of skew products with a two-dimensional phase space. The most studied at present is one of the subspaces (in some natural sense, the simplest) containing an open (but not everywhere dense in it) subset of $C^1$-smooth Omega-stable skew products. The approximation properties of such mappings are considered. It is shown how naturally, within the framework of the study of skew products, one of the possible approaches to the concept of integrability of a discrete dynamical system arises. Criteria of integrability of a discrete dynamical system are given. Unsolved problems are formulated.