2010/10/01 by Johannes Kellendonk, J. Kellendonk, Kellendonk, J. +3
Computer Science · Materials Science · Mathematics · #37b50 #47c15 #52c22 #94A55 #Cellular Automata and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Operator Algebras (math.OA) #Quasicrystal Structures and Properties #math.OA #msc:37b50 #msc:47c15 #msc:52c22 #msc:94A55
paper · pdf · doi:10.48550/arxiv.1010.0156
40 pages, 1 figure
openalex publication_date 2010/10/01 · arxiv created 2010/10/22 · arxiv updated 2010/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.