We prove that circle maps with a flat interval and degenerate geometry are an example of a dynamical system for which the topological classes don't coincide with the rigidity classes. Contrarily to all the well-known examples in one-dimensional dynamics (such as circle diffeomorphisms, unimodal interval maps at the boundary of chaos, critical circle maps) we show that the class of functions with Fibonacci rotation numbers is a C^1 manifold which is foliated with finite co dimension rigidity classes. This is a joint work with M. Martens.