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.

中文翻译：

---

几乎通勤对的重整化

在本文中，我们使用几乎交换对的形式主义给出了临界圆映射重整化的双曲性的新证明。我们将重整化扩展到环的二维耗散图，这是一维临界圆图的小扰动。最后，我们证明了位于重整化算子稳定集合中的二维映射具有拓扑上为圆的吸引子。这样的圆很关键：它上面的动力学是拓扑的，但不是平滑的，与刚性旋转共轭。