On circle maps with a flat interval and Cherry flows

We study C^2 weakly order preserving circle maps with a flat interval. We prove that, if the rotation number is of bounded type, then there is a sharp transition from the degenerate geometry to the bounded geometry depending on the degree of the singularities at the boundary of the flat interval.
The general case of functions with rotation number of unbounded type is also studied. The situation becomes more complicated due to the presence of underlying parabolic phenomena.
Moreover, the results obtained for circle maps allow us to study the dynamics of Cherry flows. In particular, we analyze their metric, ergodic and topological properties.