  We study $C^{2}$ weakly order preserving circle maps with a flat interval.
The main result of the paper is about a sharp transition from degenerate
geometry to bounded geometry depending on the degree of the singularities at
the boundary of the flat interval. We prove that the non-wandering set has zero
Hausdorff dimension in the case of degenerate geometry and it has Hausdorff
dimension strictly greater than zero in the case of bounded geometry. Our
results about circle maps allow to establish a sharp phase transition in the
dynamics of Cherry flows.
