  In this paper we give a new prove 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
demontsrate that a two-dimensional map which lies in the stable set of the
renormalization operator possesses an 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.
