Hyperbolicity of the renormalization operator for critical $\mathcal{C}^{r}$ circle mappings

We propose to study the renormalization operator acting on critical $\mathcal{C}^r$ circle mappings. (More precisely, the operator acts on critical commuting pairs.) Assuming that there is a Banach manifold of critical analytic commuting pairs on which the renormalization operator acts hyperbolically (with non-trivial hyperbolic attractor), we prove that, for r > 2, the operator remains hyperbolic with the same expanding subspaces when acting on $\mathcal{C}^r$ commuting pairs. By this we mean that the tangent renormalization operator admits a hyperbolic splitting with the same unstable subbundle.