  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.
