| Let Λ be a uniformly discrete point set in Rd with multiple colours. Then its weighted two-point correlation measure and its diffraction measure are determined by its structure. The inverse problem for Λ is to determine the structure of the point set Λ by using its weighted two-point correlation measure. However, the fact is that knowing only the two-point correlation measure usually is not enough to do this. One principal purpose of this thesis is to understand what underlies this fact.;Moreover, μ uniquely determines a stationary point process (I, Rd , μ), where X := supp(μ). In the case that μ is ergodic, we will prove that the n + 1-point correlation measure of the point process is equal to the n-th moment of the Palm measure for n = 2, 3,…. This result generalizes Gouéré's argument for the case that n = 2. Meanwhile, basing on Steven Dworkin's argument that the diffraction of typical point sets comprising X is related to the dynamical spectrum of X, we will prove that there exists an Rd -equivariant, isometric embedding that takes the L 2-space of Rd under the diffraction measure into L2( X, μ) and the algebra generated in L2 (X, μ) by the image of this embedding is dense in L2(X, μ). It will follow that the full information about μ is available from the weights and the set of all correlations (that is the two-point, three-point,…, correlations). This thesis will end with a discussion about two particular point processes.;We will show that if the frequency of every local patterns of Dm r (the space of all r-uniformly discrete m-coloured point sets) exists at Λ, then Λ uniquely determines a stationary probability measure μ on Dm r . This measure μ contains the basic information of the structure of Λ. |
