Aperiodic order and spherical diffraction, II: Translation bounded measures on homogeneous spaces

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on $\mathbb R^n$. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.Comment: Formerly part of arXiv:1602.08928. Completely revised and extended versio

Banach spaces-valued ergodic theorems and spectral approximation

The present dissertation thesis is concerned with Banach space-valued convergence theorems along Folner type sequences in geometries with amenable structures. The text contains new ergodic theorems for groups, as well as an almost-additive and a subadditive convergence theorem for hyperfinite, weakly convergent graph sequences. All these results - among them abstract mean and pointwise ergodic theorems - significantly extend the literature on ergodic theorems. As an application, one obtains the uniform convergence of the integrated density of states (IDS) for pattern-invariant, finite hopping range operators in a wide range of amenable situations. A different kind of spectral approximation is given by a strong result for Ihara's Zeta function associated with sofic graphings. It is shown that compact convergence can be obtained for the normalized finite versions corresponding to the elements in a weakly convergent graph sequence

Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions