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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.
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 measu
We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical model
We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally fin
We show that joinings of higher rank torus actions on S-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
For a large class of irreducible shift spaces $XsubsettA^{Z^d}$, with $tA$ a finite alphabet, and for absolutely summable potentials $Phi$, we prove that equilibrium measures for $Phi$ are weak Gibbs measures. In particular, for $d=1$, the result holds for irreducible sofic shifts.