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In this paper we give a new proof of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demonstrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses attractor which is topologically a circle. Such a circle is critical: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.

Renormalization of almost commuting pairs