Spectrum of dynamical systems arising from Delone sets

1998 ◽  
pp. 265-275 ◽  
Author(s):  
Boris Solomyak
Keyword(s):  
Dynamical Systems ◽  
Delone Sets

scholarly journals Delone dynamical systems and spectral convergence

2018 ◽  
Vol 40 (6) ◽  
pp. 1510-1544 ◽  
Author(s):  
SIEGFRIED BECKUS ◽  
FELIX POGORZELSKI
Keyword(s):  
Dynamical Systems ◽  
Systems Approach ◽  
Special Focus ◽  
Bounded Operators ◽  
Special Situation ◽  
Ergodic Limit ◽  
Spectral Convergence ◽  
Recent Developments ◽  
Delone Sets ◽  
Uniquely Ergodic

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

scholarly journals Equicontinuous Delone Dynamical Systems

2013 ◽  
Vol 65 (1) ◽  
pp. 149-170 ◽  
Author(s):  
Johannes Kellendonk ◽  
Daniel Lenz
Keyword(s):  
Dynamical Systems ◽  
The Other ◽  
Almost Periodic ◽  
Other Hand ◽  
Local Complexity ◽  
Dirac Comb ◽  
Delone Set ◽  
Delone Sets

AbstractWe characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with finite local complexity, the only equicontinuous systems are then shown to be the crystallographic ones. On the other hand, within the class without finite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems that are not crystallographic. Our results solve the problem posed by Lagarias as to whether a Delone set whose Dirac comb is strongly almost periodic must be crystallographic.

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