Robust rigidity for circle diffeomorphisms with singularities

We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C-1-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are C-1-smoothly conjugate.