We use a renormalization approach to demonstrate that certain dissipative perturbations of one dimensional critical circle maps admit an invariant attractor which is a non-smooth circle in an annulus.

Furthermore, 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.

This is a joint work with M. Yampolsky and D. Lilja.