C∗C∗-algebras of Penrose hyperbolic tilings

Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C∗C∗-algebras and of the KK-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C∗C∗-algebras are traceless. Nevertheless, harmonic currents give rise to 33-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling.