Simultaneous renormalization of commuting circle diffeomorphisms

We define a renormalization operator for pairs of commuting orientation preserving circle diffeomorphisms of class $C^3$. The operator is driven by the accelerated Brun multidimensional continued fraction algorithm. For rotation vectors of simultaneously bounded type, under a summable strong convergence condition satisfied by all eventually periodic types with Pisot period matrix, the renormalizations converge exponentially fast in $C^2$ to pairs of rigid rotations. The pair is simultaneously $C^{1+\gamma }$ conjugate to the pair of rotations, with $\gamma >0$ depending only on the bounded type constant. A fast renormalization extends exponential convergence and $C^1$ rigidity to a class of rotation vectors of full Lebesgue measure, containing explicit vectors outside bounded type. We show that the operator is topologically hyperbolic. The partition of the renormalized pairs by rotation vector is an invariant lamination whose leaves are the topological conjugacy classes. All results hold for pairs of class $C^{2+\alpha }$.