Pure point spectrum for measure dynamical systems on locally compact Abelian groups

Abstract

We show equivalence of pure point diffraction and pure point dynamical spectrum for measurable dynamical systems built from locally finite measures on locally compact Abelian groups. This generalizes all earlier results of this type. Our approach is based on a study of almost periodicity in a Hilbert space. It allows us to set up a perturbation theory for arbitrary equivariant measurable perturbations.

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

“…This is motivated by the following result on the autocorrelation γ (Φ) P of a given realisation, which is somewhat implicit in the literature. Its importance in our present context was first emphasised by Goueré in [28]; see also [43] for complementary aspects.…”

“…Over the past 20 years or so, considerable effort has been put into understanding the mathematics of diffraction, especially conditions under which Λ is pure point diffractive, in the sense that γ is a pure point measure, compare [32,16,55,2,8,29,42,43]. At this point in time, we have a good collection of models for producing pure point diffraction, particularly the cut and project sets (or model sets).…”

On the Pisot Substitution Conjecture

“…For example, if T is a primitive FLC substitution tiling in R n and Γ T = m i=1 Γ i × {i} is a substitution Delone multi-color set associated with T , then each Γ i has pure point diffraction spectrum if and only if (Ω T , R n ) has pure discrete spectrum ( [17,18,48,60,80,81,82,102]). …”

Aperiodic crystals and beyond

“…Three different approaches have been used to prove this statement. The first proof using methods of dynamical systems theory was completed by Schlottmann (2000), employing an argument by Dworkin (1993) and the mathematical diffraction approach of Hof (1995); see also Lenz & Strungaru (2009) for further developments. An alternative approach employs the theory of almost periodic measures (Baake & Moody, 2004;Moody & Strungaru, 2004;Strungaru, 2005).…”