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Spectral triples and characterization of aperiodic order
Author(s) -
Kellendonk J.,
Savinien J.
Publication year - 2012
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdr025
Subject(s) - mathematics , lipschitz continuity , aperiodic graph , topological conjugacy , metric space , metric (unit) , characterization (materials science) , exponent , pure mathematics , combinatorics , discrete mathematics , linguistics , operations management , materials science , philosophy , economics , nanotechnology
We construct spectral triples for compact metric spaces ( X , d ). This provides us with a new metric d̄ s 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 d̄ s and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that d̄ s 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.