Universality but no rigidity for two-dimensional perturbations of almost commuting pairs

TL;DR

This paper investigates two-dimensional perturbations of almost commuting pairs, demonstrating that while they exhibit universality and approach a normal form at small scales, they generally lack smooth conjugacy on their critical attractors, extending the 'universality but no rigidity' paradigm.

Contribution

It extends the 'universality but no rigidity' concept to two-dimensional dissipative maps near critical circle maps, showing the limits of smooth conjugacy despite universal behavior.

Findings

Maps exhibit universality and approach a normal form at small scales.

Two maps are generally not smoothly conjugate on their critical attractors.

The results extend the 'universality but no rigidity' paradigm to new dynamical systems.

Abstract

In this paper we consider two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. It has been shown earlier that such perturbations admit an attractor which is a non-smooth topolgical circle - a "critical" circle. We study conjugacies of the maps that admit such attractors and show that although the maps exhibit universality - they approach a certain normal form when looked at small scales - two maps in general can not be smoothly comjugate on their critical attractors. This result extends the paradigm of "universality but no rigidity" in two dimensions, discovered by A. De Carvalho, M. Lyubich, M. Martens, to yet another class of dynamical systems.