Point Sets and Dynamical Systems In the Autocorrelation Topology

Abstract

AbstractThis paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groupsG. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets ofGis 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 aG-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 aG-dynamical system, is a factor of the dynamical local hull.