Spectral triples and characterization of aperiodic order

Abstract

We construct spectral triples for compact metric spaces (X, d). This provides us with a new metricds 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 ofds and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove thatds 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.

A classification of aperiodic order via spectral metrics and Jarník sets

“…The main result of [39] showed that the spectral metric is Lipschitz equivalent to the underlying ultra metric if and only if the continued fraction entries of the slope of the Sturmian subshift are bounded, which in turn is equivalent to several known notions of aperiodic behaviour, as we will shortly explain in further detail. The typical choice of weightings used in [39] to define the spectral triple (in particular the Dirac operator) is suggested to be δ n = ln(n)n −t , where t > 0, and investigations of spectral metrics when δ n = n −1 have recently been carried out in [36]. Thus our choice of the weightings δ n = n −t , for t > 0, is a natural choice, generalising and extending this line of research.…”

“…While spectral triples for cross product algebras of the form C(X) σ Z seem difficult to set up, see for instance [11,19,54] and references therein, there has been a lot of activity in constructing spectral triples for commutative C * -algebras C(Y ), where Y does not carry an obvious differential structure. A series of works has been devoted to general metric spaces [5,52,53,54] and specially to sets of a fractal nature [4,6,10,20,27,32,33,36,39,40,42,43,56]. Kellendonk and Savinien [39] proposed a modification of the spectral triple and spectral metric pioneered by Bellissard and Pearson [10], which in turn stems from the work of Connes [20] and Guido and Isola [32,33], that can be used to analysis Sturmians subshifts; this construction was later generalised to minimal subshifts over a finite alphabet in [38].…”

“…Kellendonk and Savinien [39] proposed a modification of the spectral triple and spectral metric pioneered by Bellissard and Pearson [10], which in turn stems from the work of Connes [20] and Guido and Isola [32,33], that can be used to analysis Sturmians subshifts; this construction was later generalised to minimal subshifts over a finite alphabet in [38]. It is with the spectral triple and spectral metric of [39] that we will work. As is often the case, once one is lead to consider certain objects by an abstract theory (here spectral triples) and these objects turn out to be useful in another field (here aperiodic order) one finds that they can be defined ad hoc, that is, without any knowledge of the abstract theory.…”

“…The essential ingredients in the construction of [39] are an infinite weighted graph (augmented weighted tree, as introduced by Kaimanovich [37]), whose (hyperbolic) boundary is homeomorphic to the given Sturmian subshift, and the notion of a choice function, which can be seen as the noncommutative analogue of a vector in the tangent space of a Riemannian manifold. The main result of [39] showed that the spectral metric is Lipschitz equivalent to the underlying ultra metric if and only if the continued fraction entries of the slope of the Sturmian subshift are bounded, which in turn is equivalent to several known notions of aperiodic behaviour, as we will shortly explain in further detail. The typical choice of weightings used in [39] to define the spectral triple (in particular the Dirac operator) is suggested to be δ n = ln(n)n −t , where t > 0, and investigations of spectral metrics when δ n = n −1 have recently been carried out in [36].…”

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

“…However, in the special case of tiling algebras, this spectral triple essentially measured the Euclidean distance between two tilings in the groupoid used to define the C * -algebra, and ignored the substitution system. Since Bellissard and Pearson's seminal result there have been a number of papers on spectral triples of tilings, see for example [13,14,17,19]. The survey article [11] explains these constructions and their relationship to one another.…”