Renormalization of almost commuting pairs

TL;DR

This paper proves hyperbolicity of the renormalization process for critical circle maps using almost-commuting pairs, extends it to 2D dissipative maps, and shows the existence of a critical attractor topologically conjugate to a rotation.

Contribution

It introduces a new proof of hyperbolicity for renormalization of critical circle maps and extends the framework to two-dimensional dissipative maps.

Findings

Hyperbolicity of renormalization established.

Extension of renormalization to 2D dissipative maps.

Existence of a critical attractor topologically conjugate to a rotation.

Abstract

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.