SPECTRAL TRIPLES AND APERIODIC ORDER

Johannes Kellendonk (1) , Jean Savinien (1)

1 ICJ - Institut Camille Jordan

We construct spectral triples for compact metric spaces (X, d). This provides us with a new metric ¯ ds on X. We study its relation with the original metric d. When X is a subshift space, or a discrete tiling space, and d satisfies certain bounds we advocate that the property of ¯ ds and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that ¯ ds and d are Lipschitz equivalent if and only if the subshift is repulsive (or power free). For Sturmian subshifts this is equivalent to linear recurrence. For repetitive tilings we show that if their patches have equi-distributed frequencies then the two metrics are Lipschitz equivalent. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the tiling. Finally, we derive Laplace operators from the spectral triples and compare our construction with that of Pearson and Bellissard.

Domaines: Algèbres d'opérateurs [math.OA]

Johannes Kellendonk, Jean Savinien. SPECTRAL TRIPLES AND APERIODIC ORDER. Proceedings of the London Mathematical Society, 2012, 3 (1), pp.123-157. ⟨hal-00529779⟩