Skew products over subshifts of finite type naturally appear when one attempts to apply the methods of classical dynamical systems to random dynamical systems. There is also a close connection between these skew products and partially hyperbolic dynamical systems on smooth manifolds.

Even for the fiber dimension equal to one, we are far from understanding what typical skew products look like. During the last 30 years there appeared several papers studying the skew products with a circle fiber. I will talk about the case when the fiber is an interval, and fiber maps are orientation-preserving diffeomorphisms.

In the work joint with V. Kleptsyn, we developed a theorem which gives us a complete* description of the dynamics of typical step skew products (fiber map depends only on a single symbol in the base sequence). We also obtained a similar result for generic skew products using an additional assumption of partial-hyperbolic nature.

*except some subset which projects onto zero measure set in the base