Point sets and dynamical systems in the autocorrelation topology

Author
Strungaru, Nicolae
Moody, R. V.
Faculty Advisor
Date
2004
Keywords
autocorrelation topology , dynamical system , point set , compact abelian group
Abstract (summary)
This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups G . The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of G is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced. The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a G -dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a G -dynamical system, is a factor of the dynamical local hull.
Publication Information
Moody, R. V. and Strungaru, N. “Point sets and dynamical systems in the autocorrelation topology”, Canadian Mathematical Bulletin, 47, 82-99, (2004). https://doi.org/10.4153/CMB-2004-010-8
DOI
Notes
Item Type
Article
Language
English
Rights
All Rights Reserved