C*-algebras of Penrose's hyperbolic tilings

Hervé Oyono-Oyono (1, 2, 3) , Samuel Petite (4)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 PIMS - Pacific Institute for the Mathematical Sciences
3 LMAM - Laboratoire de Mathématiques et Applications de Metz
4 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

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*-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C*-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labelling.

Mots clés: C*-algebras of dynamical systems; Hyperbolic aperiodic tilings; K-theory; Cyclic cohomology

Domaines: Algèbres d'opérateurs [math.OA]; Systèmes dynamiques [math.DS]

Hervé Oyono-Oyono, Samuel Petite. C*-algebras of Penrose's hyperbolic tilings. Journal of Geometry and Physics, 2011, 61, pp.400-424. ⟨hal-00383342v2⟩